Tribute to
Akira Okubo
Portrait of Akira Okubo by Deborah Bray
mathematician with a keen sense of physics, biology, and
a rare insight into where the significant problems lay.
Actively sought out by leading marine scientists
Malcolm J. Bowman
worldwide
over tile years, Akira could identify almost
School of Environmental and Marine Sciences
100
collaborators.
As sole author and with his colThe University of Auckland, Tamaki Campus
leagues,
he
published
well over 150 papers. His own
Auckland, New Zealand
remarkable contributions earned him numerous honors, including the prestigious Medal of the
Robert E. Wilson
Oceanographical Society of Japan, and a Senior Visiting
Jeannette Yen
Scholarship at the University of Oxford. His studies
Marine Sciences Research Center
ranged from dye diffusion in the ocean, circulation in
State University of New York, Stony Brook, New York, USA
oceanic fronts, Lagrangian dispersion, and midge
swarming behavior from its chemistry to ethology. He
Akira Okubo, Professor of Mathematical Ecology at
applied his insights into turbulent mixing to subjects as
the Marine Sciences Research Center, State University
disparate as seed dispersion, animal grouping behavior,
of New York at Stony Brook, died on I February 1996, a
and spider webs.
few days before his 71st birthday after a long, couraHis text Diffusion and Ecological Problems: Mathematical
geous battle with cancer.
Models
(Springer-Verlag, 1980) remains a modern clasAkira Okubo came to the United States as a young
sic. In his preface, Akira talked about seven influential
man in 1959, following a career as Chief of the Chemical
colleagues, whom he called his seven "lucks":
Oceanography section of the Japanese Meteorological
J.G. Skellam (for encouragement to write the book);
Agency in Tokyo. He completed his Ph.D. in 1963 under
H.C. Chiang (for midge swarming
the tutelage of Donald W. Pritchard at
collaboration); J.R. Schubel (for broad
the Chesapeake Bay Institute of Johns
Trained as a chemist and
minded leadership); K. Yano and S.A.
Hopkins University in Baltimore. He
Levin (for their Japanese and English
chemical oceanographer,
continued working at CBI until 1975,
publications of a book on ecological
when he was appointed Professor of
Akira became the compleat
modeling); G. N. Parker (for translatMathematical Ecology at the Marine
theoretician...
ing the original Japanese text and
Sciences Research Center, State
later improving the English text of
University of New York, Stony Brook.
Akira's
book);
the
Hashimotos of the Hana restaurant of
He quickly established himself as a distinguished facPort
Jefferson,
New
York (for keeping Akira from starvulty member, not only in physical oceanography, but
ing);
and
to
all
his
friends
for their help and encouragealso in many other disciplines that incorporated his
ment.
Finally
Akira
thanks
Chiyo "for her companionship
wide interests in oceanic diffusion, animal and insect
in
the
long,
long
agonizing
period
of writing." Chiyo was
swarming, and studies at the physical-biological
Akira's
pet
hamster.
interface.
Akira's greatest attributes were his warm humanity,
Trained as a chemist and chemical oceanographer,
generosity
and love for science. Few scientists have
Akira became the compleat theoretician - an applied
Preface
4
Oceanography • Vol. 12 • No. 1/1999
touched as many other's lives and hearts. Through his
own enthusiasm, his generosity with his time and ideas,
and his unselfish way of allowing others to take credit
for ideas that were originally his, he multiplied his
influence amongst his colleagues and students many
times over.
On 21-22 July 1995, the Okubo Symposium was held in
his honor at the Marine Sciences Research Center of the
State University of New York, Stony Brook. The invited
speakers included Simon Levin (Princeton University),
Thomas Powell (University of California at Berkeley),
John Steele (Woods Hole Oceanographic Institution),
Mimi Koehl (University of California at Berkeley), Trevor
Platt (Bedford Institute of Oceanography), and Curtis
Ebbesmeyer (Evans-Hamilton, Inc.). Each speaker presented seminars on topics where Akira had made substantial contributions, as well as paying him a personal
tribute. Some of these participants have contributed to
this volume. Akira was profoundly grateful for the symposium and actively participated in it. By all accounts,
the symposium was a great success.
Everyone who came into contact with Akira found
their life immeasurably enriched; he became their sensei.
We present this compilation of articles and stories as a
lasting tribute to a humble, gentle man called Akira
Okubo who touched our lives in profound ways. We
have tried to capture the essence of Akira's personality
and his approach to science, his love of the inner structure and innate beauty of the physical, biological, and
ecological processes which define the natural world,
and his keen eye for the exquisite. We hope that this volume is a fitting tribute to this great scientist, humanist,
and lover of nature.
Figure 1. Malcohn Bowman paying tribute to
Akira Okubo in May 1995for his participation
in the MSRC graduate seminars over the
previous 20 years.
We wish to thank all who have contributed to this
volume in some way. And, thank you Akira for being a
treasured colleague and special friend to each of us.
Oceanography • Vol. 12 • No. 1/1999
Akira Okubo: A Man for All Seasons
and M a n y D i s c i p l i n e s
Donald W. Pritchard
Marine Sciences Research Center
State University of New York, Stony Brook, New York, USA
Jerry R. Schubel
New England Aquarium, Boston, Massachusetts, USA
and
Former Dean and Director
Marine Sciences Research Center
State University of New York, Stony Brook, New York, USA
Several other papers appearing in this volume are
tributes or describe personal experiences with this
exceptional man, Akira Okubo. Our contribution is
distinct in that between the two of us, we were close to
Akira over his full career in the United States. This started with his arrival at the Johns Hopkins University
(JHU) in 1959 and continued with his appointment at
the Marine Sciences Research Center of the State
University of N e w York at Stony Brook in 1975. We had
the opportunity to watch Akira mature and grow as a
research scholar, teacher, mentor, and colleague.
First, a little history. In the late fall of 1958 one of us
(DWP, then Director of the Chesapeake Bay Institute and
Chairman of the Department of Oceanography at JHU),
received letters from the Japan Meteorological Agency
(JMA) and the Japan Atomic Energy Agency (JAEA), asking that the Department of Oceanography accept Akira
Okubo, a chemical oceanographer and an employee of
the JMA, as a graduate student. Included with these letters were copies of publications authored by Akira.
Among Akira's publications were two papers
describing the analysis of data collected in a field study
concerning the movement and diffusion of dye patches
released into the surface layers of the ocean, using rhodamine-B dye as the tracer. These papers led to a special
interest in Akira's research. Akira and another former
student, Jim Carpenter, were both just completing their
development of improved procedures and instrumentation for measurement of fluorescence at very low concentrations of rhodamine dyes.
Following an exchange of correspondence and receipt
of Akira's formal application, a problem developed in
that the US Embassy in Tokyo refused to issue him a student visa because he had failed the required oral English
examination. Based on the earlier exchange of correspondence, including letters written in Akira's characteristic handwriting, it was clear that Akira could write
and read English adequately. Akira was at the time a
relatively mature individual to be starting a graduate
student program, being 34 years old and only 3 years
younger than his advisor Donald Pritchard. Akira was
subsequently advised to have his agency obtain a visa
for him as a scientific visitor using any available proce5
felt both awed and proud that Akira had come to JHU
dures, and the visa problem would be sorted out once he
to study turbulent diffusion in water and in just over
arrived at JHU. Somehow, this somewhat illegal procetwo years had become the world's expert in this field.
dure was pushed through the bureaucracy, and Akira
The year 1963 marked two important events. First, in
arrived in Baltimore during the summer of 1959.
January,
Akira received his Ph.D. The second was his
It is not easy to describe Akira's scientific career in
publication
of the first of a series of papers entitled
terms of a limited number of clear-cut categories. We
"Gleaning from a field of dye release
might consider that Akira had three
Akira
began
paging
experiments."
This first of the series
distinct careers, with a few short-lived
had
the
subtitle:
"1. Chaos from the
side trips in between. Akira's first sci- through a small pocket dictionary,
beginning."
This
marked the first
entific contribution, a paper "On the
pretending not to understand.
indication
that
Akira
realized the
heat of mixing of seawater," published
It suddenly became clear at that importance of chaos in oceanic diffuin 1951, was characteristic of eleven of
sion, although it would not be until
the thirteen papers he wrote prior to point that Akira really intended
the mid-1980's, during his third
coming to Baltimore. These eleven to change his career orientation.
career, that Akira actually applied
papers, extending over a nine year perithe
emerging
chaos
theory to oceanic and atmospheric
od, included descriptions of the chemical, physical and
turbulent diffusion.
biochemical properties of certain constituents of nearby
During the 12 year period 1960 through 1972, Akira
ocean waters and their related biochemical processes.
produced
37 papers, all on the subject of the physics of
These publications represented Akira's output during
diffusion
in
the ocean. This 12 year interval constitutes
his first career as a chemical oceanographer.
the
period
when
Akira was active solely in his second
As already mentioned, Akira had published two
career.
He
continued
to contribute significant new
papers during the last few years of his employment by
knowledge
on
the
physical
properties and processes of
the JMA in which an analysis of the diffusion of dye
turbulent
diffusion
in
the
ocean.
This included theoretpatches introduced into the surface layers of the ocean
ical
explanations
for
observations
of the variations in
was presented. These two papers represented the first
the
concentration
distributions
of
natural
and artificialtentative entry into his second career as a physical
ly
introduced
tracer
materials
under
various
geometric
oceanographer. Here Akira specialized in understandconstraints
a
n
d
/
o
r
various
external
driving
forces.
ing the physics of oceanic diffusion and of the developIn 1965 Akira published his first joint paper with
ment of clear, elegant theories.
Harry
Carter, starting a close association which lasted
When Akira first arrived at the JHU, he met with
for
nearly
20 years (Figure 2).
Pritchard to discuss his graduate program of course
work and thesis research. Despite the fact that, in letters
accompanying his application to JHU, Akira had
described his desire to concentrate his graduate study
on oceanic diffusion, he was still considered to be a
chemical oceanographer who had used his knowledge
of chemistry to select a dye well suited as a tracer for
oceanic diffusion. Akira was therefore asked if he had
met with Dave Carritt, the Department's senior chemical oceanographer. Akira began paging through a small
pocket dictionary, pretending not to understand. It suddenly became clear at that point that Akira really
intended to change his career orientation.
Figure 2. Professors Emeriti Donald Pritchard (left) and Harry
Carter (2nd from right) share a joyous reunion with Akira Okubo
During the two year period 1960-1961, Akira worked
and Robert Wilson (right) on the occasion of the Okubo
diligently and quietly as a student. He excelled in all of
Symposium, July 1995, held at MSRC.
his course work, but he spent most of his time analyzing data, collected for the most part by others, on the
Sometime in 1971 Akira came into Pritchard's office
temporal and spatial distribution of both natural and
and
announced that he wanted to become a biologist.
artificially introduced tracers in the surface layers of
Receiving
an expression of surprise, Akira explained
natural water bodies. With the exception of 1952 and
that
he
wanted
to study the counteracting processes of
1955, during his first career as a chemical oceanographbiological
attraction
and turbulent diffusion in insect
er, 1960 and 1961 were the only years in which Akira
swarms.
It
was
explained
to Akira that his research supdid not publish. In 1962 he showed that he had in fact
port
came
from
the
Office
of Naval Research (ONR) and
started his second career as a physical oceanographer
the Atomic Energy Commission (AEC) and that it
specializing in oceanic diffusion, with the publication of
would be difficult to justify support from these sources
three papers in this new field.
for a shift to the study of terrestrial insects. It was sugThe first of these papers can be considered as a semigested that perhaps if Akira instead studied fish schoolnal contribution. Pritchard has often remarked that he
6
Oceanogrophy • Vol. 12 • No. 1/1999
rated and at which he spent extended periods of time.
ing or zooplankton distributions, then financial support
Jerry Schubel moved to Stony Brook in 1974 as
might be justified.
Director of the Marine Sciences Research Center
Akira rejected this suggestion, saying that useful data
(MSRC). During those early years MSRC was quite
on these coupled biological - physical phenomena were
small but there was a commitment from the
only available from studies of insect swarming.
University's administration to develop a center of excelPritchard's concern was not assuaged by being informed
lence. The focus was placed on the coastal ocean and on
that the midge species to which Akira would initially be
developing scientific solutions to problems resulting
applying his theories was named Anarete pritchardi Kim.
from society's multiple and conflicting uses of the
In spite of warnings of loss of support, Akira was not
coastal zone. Akira was Schubel's first appointment to
to be dissuaded. The first three papers he published in
the faculty of MSRC in 1975.
1972 dealt with mathematical modeling of the swarmFor the first time in his career in the United States,
ing of small animals in motion. Thus began Akira's
Akira enjoyed the security of a "hard money" position.
third career. For our purpose here the short description
With total freedom to pursue any interests he chose, he
"Mathematical Biology" will serve to identify the subthrived in his third career and became
jects encompassed by Akira's third
Akira led because
a role model of a good professor for all
career.
of the power of his ideas and
his younger colleagues. Several of the
It is of interest to note that Akira did
administrators at the highest level of
ultimately find sources of data from
his ability to quietly get people
the University wondered how Akira
the marine environment which could
to work together
with his new interests fitted into the
be used in the development of models
to find solutions.
model that had been proposed, but it
describing the counteraction of biosoon became clear to everyone what a wonderful
logical attraction and turbulent diffusion. Starting in
resource he was!
1977, he published some 32 papers dealing with the
From the outset, Akira was a key element in the stratinteractions of physics and biology in the marine enviegy to build MSRC as a multi-disciplinary center. His
ronment.
collegial and nurturing nature, his capacity to formulate
Akira's first shift in careers, that of leaving behind his
complex problems of all kinds into tractable mathematresearch as a chemical oceanographer to initiate
ical forms, and his ability to find order in chaos, simresearch as a physical oceanographer specializing in the
plicity in complexity and good in everyone, ensured his
physics of turbulent diffusion, was marked by a sharp
scientific leadership in the development of the MSRC
break. After his arrival at JHU in late 1959, he never
community
of scholars, practitioners, and students.
again wrote a paper that would be identified as chemiAnd
what
a
leader he was! Akira led because of the
cal oceanography. He appeared to simply want to forget
power
of
his
ideas
and his ability to quietly get people
this phase of his life.
to
work
together
to
find solutions. Akira was the ultiThe period, beginning in 1992, which marked the
mate collaborator. He was one of the "golden threads"
beginning of his third career was quite different. During
that made MSRC such a productive learning communithis time he never abandoned his interests in the
ty. With his loss, important segments of the fabric
physics of turbulent diffusion as he was simultaneousseemed to unravel; no one has quite been able to weave
ly increasing his time and effort spent on studies of
them back together. It will come, but some time must
mathematical biology.
pass before the void is filled.
Akira did not immediately develop his bent toward
Somewhere in heaven, we suspect that Akira is snailcooperative studies and his talents for helping others.
ing and working to answer once and for all the age old
During his early years as a student and then as a staff
question of "how many angels can dance on the head of
member at the Chesapeake Bay Institute, Akira was a
a pin?"
friendly but quiet individual whose interactions were
not extensive. He gradually changed. At first his interactions were limited to just a few individuals, notably
Seeing Double - Akira Okubo and the
Harry Carter and Don Pritchard, but they continued to
Midges Anarete pritchardi
develop as he worked with and helped a growing number of individuals, including Jerry Schubel. These interAlan J. Elliott
actions can be traced by noting Akira's co-authors on
Marine Science Laboratories, University of Wales
his various publications during this period.
Bangol, Menai Bridge, United Kingdom
Spanning three separate careers, Akira spent nearly
all of his professional life in the United States at two
When I first travelled to The Johns Hopkins
institutions: the Johns Hopkins University's Chesapeake
University in Baltimore, Maryland, in 1972, it was to
Bay Institute and the State University of New York at
join the Department of Earth and Planetary Sciences
Stony Brook's Marine Sciences Research Center. During
and work with Professor Owen Phillips for a period of
these years he benefited from a rich and diverse network
a year as a post doctoral fellow on problems related to
of individuals and institutions with whom he collaboOceanography
•
Vol. 12 • No.
1/1999
7
surface waves. However, I was given an office -- not in
the same building as most of the D e p a r t m e n t - but at
the far end of the campus in the building occupied by
the Chesapeake Bay Institute (CBI). It was further suggested that I might take a good look at the Institute and
see if there were any active research projects that looked
interesting.
As a consequence I found myself settled in an office
on the ground floor of the CBI building, w e d g e d in
between Francis Bretherton and Ray Montgomery, with
Don Pritchard's office located at the end of the corridor.
Bretherton and Pritchard quickly convinced me to consider p r e p a r i n g a proposal for submission to the
National Science Foundation (NSF) on the d e v e l o p m e n t
of depth-resolving estuarine models. The proposal was
funded and led to the collection of current, sea level and
wind data over a year long period and revealed the
variability in the "classical" picture of the residual circulation within an estuary. Thus to m y good fortune, by
the end of m y post-doctoral year, with the successful
grant application to NSF, I was to spend a further three
years in Baltimore,
The CBI building was a fine example of a stratified
society. With the exception of myself the ground floor
was generally filled by full professors and the status of
the occupants on the other floors decreased as one
m o v e d upwards. The lowest members in the pecking
order (the graduate students, among w h o m were counted Alan Blumberg and Breck Owens) were grouped
together on the fourth floor. The top floor also housed
the data processing group and contained a n u m b e r of
machines that were used to generate p u n c h e d cards for
p r o g r a m m i n g and data entry. These machines were also
connected to a couple of optical/mechanical devices
that converted the data photographically stored in
current meters into digital form. The current meter digitizers were m a n n e d by a rota of casual staff hired from
a local secretarial agency.
The same room also contained in one corner a light
table at which a very quiet, slight gentleman was busily examining images projected by a light source onto a
grid of graph paper. This, I was to discover later, was
Akira Okubo, one of the finest physical oceanographers
of his generation. It was typical of Akira's gentle personality that I was not introduced to him on m y first
days in the d e p a r t m e n t but had to discover him informally at coffee time (I can also remember with wry
h u m o r that the finest office in the building - the only
one with a reclining leather chair - belonged to an
administrator whose main actions involved handing a
set of keys to new staff members. Akira was awarded no
such s u m p t u o u s office or leather chair; he was banished
to the badlands of the fourth floor where he did not
even have an office - just a table in a corner of the data
preparation room).
I spent m a n y hours on the top floor during m y first
year at Johns Hopkins, punching data and programs
onto cards, and Akira was always there bent over the
light table. I think that it was quite a while before I got
to know him sufficiently well to enquire about his activities - and I then received an account of his midge
swarming analyses. The main points are now well
known since this work is described in his extremely
readable b o o k Diffusion and Ecological Problems:
Mathematical Models (Springer-Verlag, 1980).
In essence, a white sheet was placed on the ground in
a corn field at m i d d a y and this attracted a swarm of
midges that hovered in a nearly horizontal patch at a
height of about 5 cm above the ground. The midges,
which only measured about 2 m m in length, were then
p h o t o g r a p h e d by a high speed cine camera. Akira's
industry at the light table led to individual midges
being identified on consecutive frames and their positions determined in the x-y (horizontal) plane. The data
set could then be analyzed to determine the velocity of
individual insects and the statistics of the patch and the
characteristics of the swarming behavior.
To digitize just one of these data sets must have been a
formidable task. But Akira had the tenacity to do them all
twice. This action was not entirely of his own choosing
but was a consequence of serendipity. The analysis was
intended to be two-dimensional since the midges were
k n o w n to swarm above the sheet. However, during his
analysis Akira was frequently puzzled by the manner in
which a single midge would suddenly separate into two
individuals. Conversely, two individuals sometimes
combined to form a single midge. When he realized why,
it was not good news: he had been plotting the shadows
of the midges along with the positions of their bodies!
The two merged (separated) whenever a midge landed
on (took off from) the white sheet. As a consequence his
data sets were worthless: bodies and shadows had been
confused in the digitization of the cine images. At this
stage I suspect that most of us would have given up and
m o v e d onto a new and less tedious problem.
But Akira was not like us. He realized that, by knowing the height and bearing of the sun at the time each
film was exposed, he could use the shadow of a midge
to determine not only its x-y coordinates, but also its
height above the ground. He could therefore solve the
three-dimensional problem by redigitizing his films.
The second reading of each film must have been significantly more difficult than the first because not only did
consecutive midge and shadow positions need to be
identified but each midge had to be put into a correspondence with its s h a d o w - and the two sets (midges
and shadows) had to be kept separate.
Such tasks occupied Akira during the months of 1972.
He left Johns Hopkins during 1973 and travelled to
Seattle, Washington to work on problems related to the
ecology of deer. This probably marked his formal transition from a physical oceanographer to a mathematical
ecologist.
The last time that I saw Akira was during the s u m m e r
of 1990 when he spent some months at the Mathematical
Institute of Oxford University and came to visit us for a
Oceanogrophy • VoL 12 • No. 1/1999
couple of days. He was researching the spread of the
grey squirrel in Britain and using it as a possible model
for the spread of AIDS within a hypothetical h u m a n
community. It was certainly one of the few times that
our physical oceanography group has sat through a
seminar on a largely ecological p r o b l e m - and enjoyed it.
There was, several years ago, a s u d d e n enthusiasm
for "multidisciplinary" studies within the UK. It was
rumoured that the funding agencies would look more
favourably on applications that combined, for example,
an analytical chemist with a benthic biologist with a
physical oceanographer. I was not convinced by the
argument: neither was the funding agency, for w h e n the
applications were ranked the "multidisciplinary" ones
came near the bottom of the heap. This experience suggested to me that it is not groups that can be made to be
multidisciplinary - but only very special individuals
w h o have a flair for such work.
Akira was one of the finest examples of the multidisciplinary scientist: a m a n w i t h m a n y talents and
immense physical insight. The Okubo oceanic diffusion
diagrams (graphs of the horizontal diffusivity versus
length scale in the ocean) serve as a reminder of the
legacy left to us by an extraordinary scientist.
The working hypothesis was that the fluctuations in
chlorophyll concentration were under turbulent control: the ideal result would be a Kolmogorov spectrum.
In 1971, Akira Okubo spent several weeks visiting
and working at the Bedford Institute. A n d so it happened that he was present w h e n the results from the
measurements described above were being analyzed.
At that time, the computations required for a spectral
analysis were more of an adventure than might seem
possible now. But eventually, a variance spectrum for
chlorophyll was produced and plotted on the line printer. A cursory glance showed that it was linear on logarithmic axes.
As I carried the hard-won prize back from the computer room to m y office, I met Akira in the corridor. He
looked at m y output, took it from me, laid a ruler on it
and penciled a rapid calculation of the slope.
"It is minus five thirds," he said.
We stared silently at the spectrum and at each other,
as if we had seen something that we were not meant to
see. As this drama was being played out, Akira and I
were, perhaps, the only people in the world interested
in this problem. Later, it was to create a minor growth
industry. To be able to discuss m y results with a sympathetic and competent colleague was invaluable for
me: it is a debt that can never be repaid.
P h y t o p l a n k t o n and Turbulence:
A Vignette
REFERENCE
Trevor Platt
Platt, T., 1972: Local phytoplankton abundance and
turbulence. Deep-Sea Res. I, 19, 183-187.
Ocean Sciences Division, Bedford Institute of Oceanography
Dartmouth, Nova Scotia, Canada
In 1970, I m a d e some measurements, using continuous fluorometry, of the short-term variations in chlorophyll concentration at a fixed depth on an anchor
station in the Gulf of St. Lawrence, Canada (Platt, 1972).
The idea was to compute the variance spectrum in
w a v e n u m b e r space to see if anything could be learned
about mechanisms responsible for the k n o w n heterogeneity in chlorophyll concentration at the scales accessible to the measurements (from 10 to 1000 m, under the
frozen field assumption).
Figure 3. Akira Okubo and Trevor Platt at the Okubo
Symposium, July 1995 held at MSRC (photo by Lita
Proctor).
Oceanography • Vol. 12 • No. 1/1999
Akira O k u b o and the Theory of B l o o m s
Lawrence B. Slobodkin
Department of Ecology and Evolution
State University of New York, Stony Brook, New York, USA
A PERSONAL NOTE
Akira Okubo was an elegant intellectual w h o became
part of our lives because of a combination of Jerry
Schubel's scientific good taste in hiring him to the
Marine Sciences Research Center at Stony Brook and
the remnants of feudalism that still haunt Japanese universities. By studying in the United States in his youth
and working here for several years, Akira had stepped
irreversibly out of the Daimyo system - by which a
senior Japanese professor passes employment opportunities on to his o w n students only. To reenter the system
is extremely difficult.
There are sometimes ways of getting around this Koichi Fujii, a former Stony Brook student, got a
professorship on his return to Japan but only after waiting several years and only in a new university in which
there were no Daimyo. It was our good fortune that
Japan failed to take Akira back. Later, when he was
9
not an acronym for anything, but actually has at least
clearly internationally eminent, he was almost too close
two disparate meanings. To population geneticists, and
to retirement to be hired (the mandatory retirement age
one large group of mathematical modelers, front theory
in Japanese public universities is 62 years).
Akira never really became assimilated into the
refers to the waves that are promulgated by certain
kinds of reproductive processes.
American culture. He followed a very personal code of
behavior characterized by hard work, abstemious
A new mutant initiated in a population dispersed on
habits and sensitive intelligence. His English was fluent
a flat surface may spread through the population. Its
frequency may prove highest near the edge of its range
but not spontaneous. In conversation each sentence was
a carefully prepared statement. He never drove a car.
of dispersal - producing a front that moves through the
He said that two thousand pounds of metal was too
population. Fronts of this kind also occur in terrestrial
invasions, as in the squirrel populations invading
dangerous a mass for one person to move about. His
England (which incidentally fascinated Akira).
physical regimen was inventive - before going on vacaAnother use of front theory, which is closer in spirit
tions he would increase the weight and distance he
to the theory of red tides, is the corncarried on his bicycle to get into conFronts of this kind also occur
mon observation that oceanic producdition for hikes and distance bicytivity may show a local maximum at
cling.
in terrestrial invasions,
the contact line between two discrete
He had affection for many people
as in the squirrel populations
water masses. The rationale is that
and many things. My personal sense
invading
England
(zohich
water masses differ in their chemistry,
is of the loss of a lonely, sensitive and
inciden tally fascinated Akira).
irreplaceable friend. I wish the friendso that if one of the adjacent masses is
ship could have been even closer. He
short in, say, phosphorus and the
played the Japanese chess game Shogi, but not Go. I
other in, say, usable nitrogen, the contact zone between
them, where mixing is occurring, will be a better place
play Go but not Shogi. I don't know if he found Shogi
players at Stony Brook, but I am sure I would have
for phytoplankton than either water mass. Following
enjoyed a game with him.
whale migrations has shown that their migratory patterns follow fronts of this sort; if they did not food
A SCIENTIFIC CONNECTION
would be inadequate (Philip Dustan, personal communication, 1994).
My relation with Akira was colored by references in
The first kind of front theory supplies the opportunihis delightful and immensely readable book Diffusion
ty
for mathematical theories, while the second is not as
and Ecoloy,icaI Problems: Mathematical Problenls (Springerinteresting
to mathematicians.
Verlag, 1980) to of one of my early papers (Kierstead
My
paper
with Kierstead in its original form was of
and Slobodkin, 1952). The paper described a mathematvery
little
mathematical
interest, although it became of
ical model which set the hydrographic limiting
greater
biological
interest
when it had been enlarged
conditions for the growth of a plankton bloom (phytoand
modified
by
Akira
and
others. It was built to
plankton, dinoflagellate or ciliate) in a chemically and
enhance
understanding
of
a
narrowly
defined biologiphysically suitable water mass, surrounded by water in
cal
problem
by
ignoring
what
was
then
considered the
which growth is impossible. Quite simply - the size of a
normal
approach
to
problems
of
phytoplankton.
It
water mass which is just big enough to permit increase
substituted
geometric
and
hydrodynamic
properties
for
of a plankton population is proportional to the square
biochemistry. This substitution made certain questions
root of the diffusion rate at the edges of the water mass
unanswerable
in any detailed way but in return permitdivided by the population's reproductive rate.
ted
extremely
practical conclusions about specific
The lower the diffusion rate and the higher the reproplankton
blooms.
ductive rate, the smaller the minimum water mass size
Writing in a memorial volume to Akira Okubo
required for a bloom. The idea was derived from critical
provides
an almost unique opportunity to expand on this
mass theory of nuclear chain reactions. The English
distinction
and its significance. It is not merely a review
statistician, Skellam, simultaneously published a small
of
a
problem
that vanished decades ago, but rather is still
book in which he described the spatial pattern of a
current
as
can
be seen in recent studies of brown tides in
freely growing population (Skellam, 1951). The two
estuaries
and
front
problems in the open sea.
systems were effectively equivalent.
THE SIGNIFICANCE OF KISS
THE ORIGIN OF KISS
Since Akira managed only with difficulty to
pronounce my last name, he constructed the fine
acronym KISS standing for Kierstead, Slobodkin and
Skellam as the name for this sort of model. Later this
became part of what may be called front theory.
Front theory superficially means what it says and is
I finished my Ph.D. in 1951, having completed a laboratory study of Daphnia population dynamics. I was
immediately appointed Chief of Red Tide Investigation
for the US Fish and Wildlife Service. I had never
cultured a phytoplankter (other than Chlanntdl/mOllaS
which I raised to feed to Daphnia) nor had I studied any
10
Oceanography • VoL 12 • No. 1/1999
biochemistry or plant physiology at all. I still believe
my appointment occurred because the mystery of the
red tide was a matter of some public relations importance to the Fish and Wildlife Service and they had no
desire to see that mystery dispelled.
Prior to 1952, the traditional way to study any planktonic organism was to identify it and if possible,
characterize in what ways it was unique. For microorganisms, one proceeded to examine the organisms in
culture tubes, following the model of medical bacteriology. By altering physical and chemical circumstances
one could make the organisms grow, encyst, or die. In
the case of pathogens of humans, the conditions which
caused these alterations could be considered as indicative of potential methods of curing patients infected
with the organisms.
In limnology and oceanography, pathogenicity was
usually of less significance. Phytoplankton and water
chemistry were studied from the standpoint of biological productivity. Photosynthetic micro-organisms lived
in culture media which were generally simple compared to the requirements of bacteria. It was of course
difficult enough to get the precise mixture of nutrient
chemicals, temperature, light regimen and trace
elements (and as was later discovered, trace organics
and vitamins) to make a phytoplankter grow.
Nevertheless, skilled hard work could make it happen.
Once the isolation, identification, and growth conditions of a particular organism were established it was
quite reasonably hypothesized that when that phytoplankter was found in nature the same, or similar,
conditions were being fulfilled in nature.
This investigative process provided fundamental
information on physiology and biochemistry for many
organisms. It took on special meaning when concern was
focused on blooms, such as red or brown tides. By extension from medical microbiology, it would seem that the
organisms ought to grow in nature under the same conditions that permitted them to be cultured in the laboratory, and conversely, altering conditions in nature so as to
make growth or survival impossible was presumably
one way of dealing with toxic and nuisance blooms.
While efforts in the laboratory focused on physiological properties of the organisms, the overwhelming
majority of field studies, at that time, consisted of what
were called surveys. The classic wisdom of biological
oceanography, going back to the 19th century, stated
that pure water and sodium chloride were obviously
inadequate media for algal growth. Phytoplankton
required nutrient salts - usually called nutrients. It was
generally understood that there were probably a fair
number of these and they had to be present in the
correct concentrations.
Phosphorus as dissolved and particulate phosphate,
nitrogen as nitrate, nitrite and ammonia and dissolved
oxygen and sodium chloride were the usual materials
analyzed in each sample of seawater. The state of analytic chemistry was such that it was impossible to
Oceanography • VoL 12 • No. 1/1999
analyze samples as quickly as they could be collected.
Therefore, an}, normal oceanographic laboratory had a
vast collection of stored water samples waiting for
ultimate analysis.
The Red Tide Laboratory of the Fish and Wildlife
Service was located in Sarasota on the west coast of
Florida. It had been established in response to the popular outcry that accompanied the red tide outbreak of
1947, which occurred in Miami on the east coast of
Florida. After the establishment of the laboratory, red
tide outbreaks did occur on the Florida west coast - but
not with the same species that had been found in the
Miami bloom.
The laboratory had a research vessel approximately
twenty five meters long with a crew of three (including
one sea-cook), two chemists, a biologist and a secretary.
I was 23 years old and the laboratory director had never
done any oceanographic research. Before I arrived, the}'
had established a pattern of cruising along the coast,
collecting samples for chemical analysis and microscopic examination of phytoplankton and zooplankton, plus
storing them in a warehouse.
Microscopic examination of the water when there
was no active bloom showed none of the bloom organisms at all. Underlying the chemical analyses was the
tacit assumption that when chemical conditions of the
water changed, they more or less closely approximated
the optimal culture conditions for one or more of tile
phytoplankters in that water which then increased in
abundance. The fact that when the bloom occurred, it
was of an organism that was not present in the normal
sea water was interesting.
Also, the nutrient concentrations of the normal seawater were very low and did not change very much
with time and certainly did not change in advance of
occurrences of blooms. It was known that chemical
enrichment of surface waters could occur by upwelling
of richer deep waters along continental shelves.
However, the continental shelf on the west coast of
Florida was about one hundred and eight}, kilometers
out to sea and there was no evidence that upwelling
was significant (I believe that more recently some signs
of upwelled water reaching the coast have been found).
When red tides occurred, the total nitrogen and phosphorus in the water was orders of magnitude higher
than that found in non-red tide water.
When red tide organisms had been collected during
the blooms, and attempts made to culture them in seawater, they died. No serious attempts were being made
at the Sarasota Laboratory to culture non-red tide
organisms.
In short, the Florida red tides were being produced
by organisms that did not occur in the plankton, in
water that did not normally exist, laden with nutrients
that could not have arrived by upwelling.
The situation was not encouraging, although the
amenities were being observed by maintaining a sampiing program and keeping the warehouse full of as vet
11
unanalyzed samples. The citizens of the town could see
that there was a busy laboratory on the main pier.
Congress knew that the "Mystery of the Red Tide" was
the subject of research and a kind of stalemate had been
reached.
The paradoxical nature of these red tides disappeared
when I focused on hydrography rather than water
chemistry. The argument of a preliminary theory of red
tides ran thus:
1. If the red tide cannot occur in normal Gulf of Mexico
water, it must be occurring in abnormal Gulf of
Mexico water.
2. By definition abnormal water occurs in smaller
quantities than normal water and if the two kinds of
water mixed, the red tide would be impossible.
3. In the absence of solid barriers, sharp density
gradients resist the forces that mix water masses. I
tentatively assumed that the abnormal water had a
significantly different density from the normal.
4. Temperature and salinity differences are the major
components of density differences between water
masses in nature. Blooms should therefore be associated with either high or low salinity water masses.
5. Since Gulf of Mexico water is already of unusually
high salinity, I guessed that the low salinity waters
would be significant in the Florida blooms. When
John Howell, the laboratory biologist and I examined
the bayous inland from the shoreline, we found that
the red tide dinoflagellates were normal inhabitants
of the brackish pools.
6. The historical record, scientific literature during the
twentieth century and newspaper accounts during
the nineteenth, showed that the red tides on the
coast of Florida were often associated with a recognizable pattern in which a relatively dry period was
followed by very heavy rain, which was in turn followed by a wind and tide pattern which might be
expected to send a pool of brackish water, containing
the dinoflagellates, into the Gulf (Slobodkin, 1952).
Henry Stommel, a brilliant physical oceanographer
who spent most of his career at the Woods Hole
Oceanographic Institution, listened to the story and put
me in touch with Henry Kierstead. Between us we
developed a more formal model (Kierstead and
Slobodkin, 1953).
Luigi Provasoli, at the Haskins Laboratory in New
York, took our samples and mud from the bayous and
succeeded in culturing the red tide organisms after
diluting the seawater with approximately 10% fresh
water. The essential ingredient in the mud was later
12
shown to be vitamin B12. Of course the abnormal, viz.,
bayou water has a reasonable amount of nutrients for
the beginnings of the bloom, and once the first fish dies
phosphorus and nitrogen are abundant.
During the next red tide we used the brand new seawater conductivity sensors aboard the MV Alaska
(which apparently spent its life in the Gulf of Mexico) to
cruise repeatedly through the edges of the colored
water. Despite peculiarities of wiring and slight
malfunction, it was immediately obvious that the red
tide was in a discrete water mass differentiated by its
salinity. Feeling vindicated, I immediately telephoned
my superiors in Washington. I enthusiastically announced to them that the mystery of the red tide had
been partially dispelled. They were not amused.
I was told to stop this sort of theorizing, to concentrate on the sample collecting cruises, phytoplankton
examination of Gulf water and on the creation of a synthetic medium for raising the dinoflagellates. I was also
told not to publish. The following morning I took my
manuscripts of the two papers I was working on and
quit my first post-doctoral job.
While the general theory was qualitatively correct, it
was distressingly anecdotal and incomplete. I was
thinking in terms of a small inoculum into an appropriate bubble, which then had to survive for around twenty days to permit bloom concentrations to develop. It
has since been demonstrated in Florida, the Gulf of
Maine, Milford Harbor (Connecticut) and elsewhere,
that many red tide dinoflagellates have an elaborate
system of alteration of generations which produces a
heavy population of spores on the sea bottom. When a
chemically and physically appropriate water mass
moves over the spore bed there can be a mass emergence, very quickly producing a red tide.
There is now a vast literature on red tides, summarized in excellent symposium volumes (e.g., Taylor and
Seliger, 1979; Cosper, Bricelj and Carpenter, 1989) and
the symposia continue (e.g., Scottish Association for
Marine Sciences, 1996).
W H Y W E R E THE FISH A N D
WILDLIFE OFFICIALS U N H A P P Y ?
Over the years I have been mildly resentful of what I
saw as the Fish and Wildlife people's lack of perspicacity. Recently I have come to understand their problem a
little better. Aside from their mendacity in wanting to
preserve "The Mystery" with minimum disturbance or
effort, they also had a clear goal which my attitude did
not address. I was concerned with the paradoxical character of the red tides and therefore focused on how they
might originate. The currently accepted story of the origin of red tides is essentially as I have described it.
Brown tides, now a chronic plague of Long Island's
Great South and Peconic Bays seem to need high salinity water, rather than low salinity but it still must occur
in stable water masses. The basic expectation of an
Oceanography • Vol. 12 • No. 1/1999
KISS THEORY IN THE CONTEXT
isolated water mass producing a bloom stays pretty
OF ECOLOGICAL MODELING
much the same.'
There is an easily verified phenomenon in which a
The KISS theory entered the main stream of ecologibottle of seawater or brackish water is loosely stopcal mathematics as Akira Okubo developed it from the
pered and kept at constant conditions for a fortnight. I
minimalistic formulations of Skellam, Kierstead and
believe that usually such a bottle will develop a bloom
Slobodkin (Akira affectionately called KISS the "Keep It
of something!
Simple Stupid" theory; a philosophy he carried into all
But, by analogy with medicine and with terrestrial
his research). Minimalistic theories typically deviate
agricultural pests, the research emphasis had, for the
from past theories by shifting the emphasis of research
last seventy years, been on "curing" the "sick" water by
to a new direction and sacrificing realistic details by
adcling something to the bloom to make the toxic organfocusing on just a few aspects of a particular problem
-',sms vanish or become harmless. It has been seriously
(Slobodkin, 1994).
suggested many times that copper sulfate, which in
Specifically, as presented by Kierstead and
proper dilution kills algae in fish tanks without killing
Slobodkin, population growth is reduced to a single
the fish, might be added to red tide. Occasionally it has
number, a growth (cell division) rate assumed constant
even been suggested that large helicopters might stir
during the course of early bloom growth.
the water in a way that would make the red tide disapThe rate of diffusion is also represented by a single
pear. Also, anti-fouling paints have also been suggested,
number without giving consideration
since they are successful at eliminatto the very difficult problem of how
Akira affectionately
ing many plankton organisms. But
turbulent diffusion can be measured
paints are not mixed into the water
called KISS the
and parameterized in particular
but form a toxic micro-layer on a hard
"'Keep It Simple Stupid" theory; cases. The size of the water mass is
substrate.
taken as a linear dimension, regarda philosophy he carried into
I can imagine nothing short of alterless of its shape. Also, no mention is
all his research.
ing water flow patterns that could
made of the initial distribution of
possibly prevent red tides. ~
organisms in the water mass. There was a mathematiI find it impossible to imagine any non-living subcal analysis in Kierstead and Slobodkin's paper which
stance, however innocuous, that could be added in any
demonstrated that after sufficient time had elapsed, the
practical way to ameliorate the effects of an existing red
proportionality constant in the equation was of the
tide. The recent silly experiments on adding iron to the
order of pi (3.14) regardless of the shape of the water
ocean to control CO-~ concentrations in the atmosphere
mass or the initial distribution of the organisms.
demonstrated that this kind of thinking still persists.
The equation therefore was of extremely limited
Even such innocuous materials as milk, sugar or sand,
predictive use other than to alert investigators to the
if used in appropriate quantities to make a serious
apparent fact that some kind of discrete water mass is
difference to a water mass, would each present a major
involved and to give some estimate of the possible minenvironmental impact.
imal sizes of the water mass - of the order of hundreds
Does this imply that there is no possible remediation
of meters. It did not predict the occurrence of cysts in
for biological phenomena that occur on the scale of the
bloom organisms.
red tides (or for example, terrestrial phenomena such as
The result specifically said nothing about the pattern
the spread of squirrels, which was also studied by
of population growth in the bloom, or the physiological
Akira)?
changes that occurred during the bloom. Finally, the
What seems necessary is a procedure which is highly
water mass was tacitly assumed to have a clear outline,
specific for the organisms concerned, so that there is no
from the standpoint of the life or death of the organisms,
collateral impact, a procedure which can be designed to
even though the border was defined by turbulent diffube applied as widespread as necessary to complete the
job, but which need not be introduced in massive initial
quantities. These criteria are just those that help define
JFor example the New Y~rk Times fry September 19, 1996 (p.16) carried an
an excellent weapon in biological warfare (Rosebury,
account of a red tide near Port Aransas, Texas in which state biolossts
1947)! ~
described it as a "typical red tide" with raill water and nutrients.h~rming a
Thus a viral or bacterial infective agent may be the
water mass containing the tide.
only practical way to "cure" a red tide (Slobodkin, 1989).
The usual pattern of seashore development in which the shore is developed
in this context it is particularly gratifying that viral
~r houses, docks and bridges, will impede water flow and tend to increase
infections of natural populations of brown tide organthe h'cquency qf red tides.
isms have recently been found. These are contagious,
~BiologicaI wapi~m' weapons may be designed to kill crops or livestock but !f
vary in virulence and may actually eventually serve as
they arc aimed at a human population they ideally do not kill but ,lelvly
a brown tide cure (Benmayor, 1996).
incapacitate temporarily. Ay,ents which destroy bloom organisms shotdd be
lethal for those organisms and otherwise harmless.
Oceonogrophy • Vol. 12 • No. 1/1999
13
sion and therefore should have been either a gradient or
a statistical entity.
There is n o w a large literature on mathematical
ramifications and complications of the minimalist
theory, starting with O k u b o himself (1980) and continuing t h r o u g h the m a n y volumes of the Springer Verlag
series: Notes in Biomathematics. A critique of these studies is well b e y o n d what is suitable for this volume,
except to note the i m p o r t a n t role of Akira in initiating
and stimulating them. It is of interest that he was one
of the very few investigators w h o was a key figure in
all the different aspects of the study of diffusion in
ecology.
In conclusion, Akira Okubo's sense of h o w mathematics can contribute to understanding the interactions
between physical processes and the spatial properties of
biological systems has generated a major, varied and
fascinating stream of intellectual effort which will be his
lasting m o n u m e n t even after his m a n y friends and
fellow travelers have passed away.
W i l d e b e e s t a n d the M a r i n e
E n v i r o n m e n t : G n u s from the Front
Simon A. Levin
Department of Ecology and Evolutionary Biology
Princeton University
Princeton, New Jersey, LISA
I N T R OD U C T I O N
It may seem strange to have a paper on wildebeest
appear in Oceanography, but it is quite logical in an issue
dedicated to the m e m o r y of Akira Okubo. Akira, after
all, saw no reason to create theoretical boundaries
between the terrestrial and marine environments; and it
was a tribute to the Marine Sciences Research Center at
Stony Brook that he was able to pursue his research on
a wide range of organisms in a diversity of habitats,
within a d e p a r t m e n t whose m a n d a t e normally would
seem m u c h narrower. It is fitting, therefore, that
Oceanography takes an equally enlightened view.
Akira first b r o u g h t theoretical attention to wildeREFERENCES
beest as a m o d e l system for s t u d y i n g patterns of
Benmayor, S., 1996: Environmental conditions influencing
animal aggregation. Wildebeest form
viral-algal interactions in the brown
a d i v e r s i t y of spatial patterns,
tide picoplankter Aureococcus anophagetIt would be a shame, though,
d
e p e n d i n g on conditions, r a n g i n g
tereus. M.S. thesis, Marine Sciences
to have his ideas disappeat,
from
long linear files d u r i n g migraResearch Center, State University New
since
undoubtedly
tion,
to
striking wave-front patterns
York, Stony Brook.
d
u
r
i
n
g
foraging.
Cosper, E.M., V.M. Bricelj and E.J.
they will stimulate others
The m e c h a n i s m s controlling
Carpenter,
eds.,
1 9 8 9 : Novel
to go on to demonstrate
Phytoplankton Blooms: Causes and
these patterns involve environmental
their richness.
impacts of recurrent brown tides and
factors, resource limitation, and interother unusual blooms. In Coastal and
individual dynamics; but in general,
Estuarine Studies, 35. Springer-Verlag, Berlin.
they all appear to be mediated entirely at local scales. In
Kierstead H. and L. Slobodkin, 1953: The size of water masses
this way, wildebeest seem analogous to other terrestrial
containing plankton blooms. J. Mar. Res., 12, 141-147.
mammals, as well as to birds, insects, fish, marine
Okubo, A., 1980: Diffusion and Ecological Problems: Mathematical
invertebrates, and a variety of other organisms. Thus
Models. Springer-Verlag, Berlin.
they provide a superb model system, and one that
Rosebur}; T., 1947: Bacterial Warfare: A critical anait/sis of the
through Akira's influence has attracted much theoretiavailable agents and the means for protection against them.
cal attention.
William's and Wiliness, Baltimore.
In my files is a long letter in Akira's fine hand, dated
Skellam, J., 1951: Random dispersal in theoretical populations.
Biometrics, 38, 196-218.
4 December 1980, in which he writes:
Slobodkin, L., 1952: A possible initial condition for red tides
Enclosed is a note on the wildebeest problem. Sorry that
on the coast of Florida. I. Mar. Res., 12, 148-155.
I
have no time for making a typed copy (leaving for
Slobodkin, L., 1989: The null case of the paradox of the plankMinnesota
tomorrow morning). It is only the begimling.
ton. In: Novel Phytoplankton Blooms: Causes and hnpacts of
I
hope
that
you
can continue~'om tlre point I stopped, or
Recurrent Brown Tides and other Unusual Blooms. E.M.
Cosper, V.M. Bricelj and E.J. Carpenter, eds., Springereven make a deviation into tire right direction.
Verlag, Berlin.
Akira and I never finished this project; we both went
Slobodkin, L., 1994: Simplicity and Complexity in Games of the
on
to other, c o m p l e m e n t a r y approaches to problems of
Intellect. Harvard Paperbacks, Harvard University Press,
aggregation. It would be a shame, though, to have his
Cambridge, Massachusetts.
ideas disappear, since u n d o u b t e d l y they will stimulate
Taylor, D.L and H.H. Seliger, 1979: Toxic Dinoflagellate Blooms.
Elsevier/North Holland, New York.
others to go on to demonstrate their richness.
14
Oceonogrophy • Vol. 12 • No. 1/1999
O K U B O ' S WILDEBEEST MODEL:
I will describe here Akira's ideas on modeling wildebeest front patterns. Technically, some of these ideas
were jointly derived; yet collaboration with Akira was
often a case of being m a d e to feel that you had thought
up things that Akira had worked out long before. Along
these lines, much could be learned from the final lines
of Akira's letter to me
p.s. I bring some 35ram slides f~r animal swarming. If
you want to use them in your talk, it is welcome. A.
In Akira's approach, patterns arose from an interaction between wildebeest and their resource (grass).
Equations were derived for the change in the number of
animals per unit area and the density of the resource.
Local wildebeest density varied in response to the animals' tendency to follow resource gradients, and to diffuse (it would have been irregular to have an Okubo
paper without diffusion); grass disappeared due to
wildebeest grazing. This leads to a quite standard formulation. Akira also proposed an alternative equation
for P, the density of wildebeest.
'~ +V
VS+bVP
"--fi'V2e 2,
Its form is preserved here as a testimonial to Akira,
and is henceforth to be k n o w n as the Okubo equation.
It involves more complicated dependence of m o v e m e n t
on the number of predators P per unit resource S, rather
than simply on the absolute amounts of each variable; a,
b and [3 represent constants.
Analysis proceeded along standard lines, looking for
traveling waves, finding fronts, and analyzing their
stability to small perturbations. It was very m u c h a
work in progress, and remains so today.
INDIVIDUAL-BASED MODELS:
Inspired in large part by Akira's influence, a variety of
other approaches have developed for explaining patterns
in wildebeest (Gueron and Levin, 1993; Gueron et al.,
1996; Chao and Levin, 1998), fish and marine invertebrates (Gr/inbaum, 1994; H u m s t o n et al., in press). These
efforts share the c o m m o n features of focusing on
individuals, and writing equations for how their movements respond to local environmental cues, including the
positions or velocities of their neighbors. From such
Lagrangian representations, one can hopefully derive
macroscopic population statistical mechanics, proceeding where possible to Eulerian descriptions, as in fluid
mechanics. Akira was a strong advocate of such mechanistic approaches to deriving diffusion approximations,
as is clear from a reading of his classic text (Okubo, 1980).
In general, the theoretical literature has focused on
the self-organizing aspects of patterns for these
systems, ignoring both environmental factors and even
the consumer-resource dynamics implicit in Okubo's
Oceanography • VoL 12 • No. 1/1999
approach. This is despite the demonstration (Segel and
Jackson, 1972; Levin, 1974; Levin and Segel, 1976;
Okubo, 1974) that such trophic interactions could give
rise to patchiness. Indeed, in some cases it is because the
patterns of patchiness in nature clearly do not match
those generated by such models, and only individual
interactions seem capable of explaining the observed
patterns (GrLinbaum, 1992).
In general, patchiness has different explanations on
different scales, and models are needed that combine
and compare the various influences. Little such work
has taken place for wildebeest; but in the marine environment, interfacing fluid dynamics, environmental
factors and inter-individual interactions has been a theoretical gold mine. Examples of such work are those of
H o f m a n n (1988) for krill, H u m s t o n et al. (in press) for
tuna, and Flierl et al. (1998) for a variety of organisms.
In all of this, the shining star of Akira Okubo burns
brightly overhead, showing the way.
A CKNO WLEDGMENT:
I am pleased to acknowledge the support of the
University Research Initiative Program of the Office of
Naval Research through grant number ONR-URIPN00014-92-J-1527 to Woods Hole Oceanographic
Institution.
REFERENCES:
Chao, D., and S.A. Levin, 1998: A simulation of herding
behavior: The emergence of large-scale phenomena from
local interactions. In: International Co~{ference on D!fiferential
Equations with Applications to Biology, 1997 Hahfax, Nova
Scotia. Fields Institute Communications Series, American
Mathematical Society.
Flierl, G., D. Gr/inbaum, S.A. Levin and D. Olson, 1998:
Individual-based perspectives on grouping. J. Theoretical
Biology.
Grfnbaum, D., 1992: Local processes and global patterns:
Biomathematical models of bryozoan feeding currents and
density dependent aggregations in Antarctic krill. Ph.D.
Thesis, Cornell University, Ithaca, NY.
Gr/inbaum, D., 1994: Translating stochastic density-dependent individual behavior with sensor}, constraints to an
Eulerian model of animal swarming. J. Mathematical
Biology, 33, 139-161.
Gueron, S. and S.A. Levin, 1993: Self-organization of front patterns in large wildebeest herds. J. Theoretical Biology, 165,
541-552.
Gueron, S., S.A. Levin and D.I. Rubenstein, 1996: The dynamics of mammalian herds: From individuals to aggregations.
J. Theoretical Biology, 182, 85-98.
Hofmann, E.E., 1988: Plankton dynamics on the outer southeastern U.S. continental shelf. Part III: A coupled physicalbiological model. ]. Marine Research, 46, 919-946.
Humston, R., J. Ault, M. Latcavage, and D. Olson. Large scale
schooling and the migration of large pelagics relative to
environmental clues. In press.
Levin, S.A., 1974: Dispersion and population interactions.
American Naturalist, 108, 207-228.
Levin, S.A. and L.A. Segel, 1976: A hypothesis for the origin of
planktonic patchiness. Nature, 259 (5545): 659.
15
Okubo, A., 1974: Diffusion-induced stability in model ecosystems.
Chesapeake Bay Institute Technical Report 86, The Johns
Hopkins University, Baltimore.
Okubo, A., 1980: Diffusion and Ecological Problems: Mathematical
Models. Springer-Verlag, Berlin.
Segel, L.A. and J.L. Jackson, 1972: Dissipative structure: an
explanation and an ecological example. J. Theoretical
Biology, 37, 545-559.
was always easy to accept his model equations, often
without appreciating the physical insight which had
gone into the simplifying assumptions. The clarity of
his logic was always so helpful and his analytical
solutions could point to the controlling parameters or
combinations of parameters, and the dependence on
initial and boundary conditions.
Akira O k u b o as C o l l e a g u e and Friend
~-- 0 ~-~,-6 ~
Robert
E. W i l s o n
o_
Marine Sciences Research Center
State University of N e w York
Stony Brook, N e w York, U S A
-7~,- A.~_. . . .
d---
.~.
el)
5:,
I want to relate some of my interactions with Akira in
February and March 1995 before his health deteriorated. I wish to do this because they were so typical and
because I have no doubt that the same types of interactions have been shared by many of his colleagues. They
show Akira's generosity and the enthusiasm that he
could bring to a problem. They also emphasize that
Akira was a true modeller with an amazing facility for
producing a quantitative framework which could be
used to develop a problem.
Akira and I had discussed time series observations of
water column temperature observations from a mooring in western Long Island Sound (Figure 4). The temperature difference between measurements taken at different depths in the water column exhibited large fortnightly and longer period fluctuations, presumably
related to variations in the intensity of vertical mixing.
We discussed whether these observations could provide
quantitative information on this mixing associated with
wind and tidal forcing.
14
14
13
13
12
12
11
11
10
10
J
-XC
t
5. L..L
i,~s(,.s., - s . )
,,~
d.S:
$ <~ ~,l./...~
~<,; :
~l~+; .
:
+.
<r;
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e"
+
<,', d ~'~" ,,.
~°
i
-
- atl
K~ts
¢¢/
J.
~:
-:dr
~:,t, . ~ & r ) ,~"
~{~): d', e
&-¢~.- k
+ Q.
<,°,/,.
~ ,o I~,.
,', "
sl.w/sv~r./;v
. ~,
~ ~'i
~..2;-.'d-*~- )
+~
(/_~,It)
( "c. O(,~vU
?
o
E
9
9
8
~4o
145
Juhan
15o
155
~6o
Days
Figure 4. Time seriesfor water column tempen#ure at 4m and
16m at NOS Station F3 (73.18" W; 41.05<'N) in western Long
Island Sound.
Following our discussion, Akira produced overnight
a first model which distilled our basic ideas. It was
based on heat conservation for each of two layers. It
16
Figure 5. Okubo's refined two-layer analytical mMel, formulated while
working oll the "Wilson Problem"for vertical heat exchange in Long Island
Sound, dated 9 February 1995.
After some discussion of this first model, Akira
produced a refined model the next evening (Figure 5). It
was mathematically more elegant and captured additional physics. Following the analytical solution there is
his typical helpful discussion of the orders of magnitude of parameters involved. He never left you a
solution without some helpful interpretations.
Akira produced a second refinement to this model
which I have not shown here. Akira was very skillful in
his abilities to make you feel as though the ideas were
Oceonogrophy • Vol. 12 • No. 1 / 1 9 9 9
..'z ~,
not so much his, just his articulation of your ideas. That
is one thing that ingratiated him to those he interacted
with. Friends and colleagues like Akira are very rare. I
am sure many friends and colleagues, like me, keenly
feel his absence.
Figure 7. Sea mine from Japan
washed ashore at Westport, Washington, during 1948-49 (photoby
Brady Engvall).
P u m i c e and M i n e s Afloat On The Sea
Curtis C. Ebbesmeyer
Evans-Hamilton, Inc.
Seattle, Washington, USA
O R I G I N OF LIFE I N FLOATING ROCKS?
with a computer simulation by
W. James Ingraham, Jr.
National Oceanic and Atmospheric Administration
National Marine Fisheries Service
Seattle, Washington, USA
When distinguished scientists die before their time,
many investigations invariably remain unfinished. In
this memorial issue I will describe two of these endeavors with the hope that Akira's ideas will live on in the
minds of young scientists.
I first spoke with Akira over the phone in 1969 concerning Snarks, the underwater clouds of anomalously
warm water drifting through Dabob Bay, one of Puget
Sound's inlets. As the years passed Akira made 50 trips
to Seattle from the Marine Sciences Research Center at
Stony Brook funded to study physical and biological
diffusion processes.
Figure6. Graypumicerockfound
floating near Antarctica in the
1960s by Mike O'Connell. in tank
tests, it floatedfor almost exactly
one year (photo by Nancy Hines,
University of Washington).
In the off-hours we discussed topics centered around
all manner of things flying and adrift, including the
accelerations of swarming mosquitoes (Okubo et al.,
1977), iceberg drift modeled like the migration of large
animals (Ebbesmeyer et al., 1980) and a bottle that drifted from China to North America containing a plea for
the release of Wei Jingsheng, China's famous dissident
(Ebbesmeyer et al., 1993).
These and other studies we completed. However, two
of Akira's favorite topics went unpublished: the origin
of life in floating rocks (Figure 6) and the drift of World
War II submarine mines across the North Pacific Ocean
(Figure 7).
Oceanography • Voi 12 • No. 1/1999
Beginning in the 1970's Akira and I began studying
many floating things of historic and scientific interest.
Objects drifting on the ocean's surface are of considerable importance to the evolution of life in general and
man in particular, but this flotsam has received little
attention, and pumice far less. We theorized that life
originated in floating pumice. We believed that publishing this idea would lead to lively and productive
debates, thereby clarifying the conditions under which
life on earth originated.
On July 25th, 1993, we submitted the following letter
to Nature magazine. Nature never published our idea, so
I take the liberty here to publish our theory in hopes
that the debate will live on in our memories of Akira.
Dear Editor:
In the earth's early histoD; volcanic activity led
to floating pumice as soon as the primordial ocean
began forming. It is well known that volcanic eruptions typically result in large quantities of pumice
and that pumice can float for years on the ocean.
Furthermore, pumice, being a matrix primarily of
voids and silicon dioxide, adsorbs a variety of
chemicals as it floats around. Given that complex
chemicals led to the origin of life on earth, and that
pumice must have been present long before these
chemicals evolved, it occurred to us that pumice
drifting on primordial seas would have been locations where complex chemicals formed and concentrated. Therefore, they may have been the
places where life originated.
As the earth cooled and primordial seas
formed, extensive volcanic activity may have
resulted in pumice covering a large fraction of
oceans that were much smaller than at present.
Chemicals adsorbed to pumice interstices
undoubtedly would have been exposed to high
concentrations of energy via radiation from the
sun and lightning strikes directly to individual
pumice stones. After the pumice became stranded
on surrounding land, evaporation and resultant
concentration of adsorbed chemicals may have
further aided in the formation of microbes.
17
Considerations of the earliest life forms evident
in the sedimentary records suggest that diversity
of life was rather great in those times. Given that
the development of diversity requires sheltered or
segregated environments, one of the most sheltered environments would have been the interstices in drifting/stranded pumice. We therefore
envision microbes evolving in the various interstices and locations on the early seas. As soon as
microbes left the protected pumice interstices,
competition could begin in the turbulent ocean,
thereby reducing diversity. If our speculation is
correct, some corollaries should be explored.
Similar processes may be continuing at present.
Pumice now drifting could be examined for the
presence of adsorbed complex chemicals. Recovery
of ancient pumice may reveal, with modern techniques, the presence of complex chemicals and
microbes, much like the preservation of insects in
amber. Useful laboratory experiments have been
conducted, for instance, by applying lightning to
sea water in the presence of ancient atmospheres.
We suggest that the addition of pumice and
some oceanographic processes may provide interesting experiments. In the primordial seas, windgenerated waves undoubtedly produced foam
which collected in windrows. In turn, the
windrows also would have collected the drifting
pumice, yielding a mixture of pumice and froth.
The froth seems a realistic location for complex
chemicals which in turn would find sheltered
environments in the pumice interstices. Lightning
striking pumice in this situation might have
sparked the formation of complex amino acids.
The structure of pumice is undoubtedly fractal in
nature (i.e. the effective surface area is very large). It
is well known that chemical reactions proceed more
rapidly on surfaces than in free water. Black pumice
absorbs heat and promotes chemical reactions.
Therefore, complex chemicals would be expected to
evolve more rapidly within pumice than in the
fluid ocean. As microbes evolved, undoubtedly
some took advantage of sunlight. Cross sections of
ancient pumice may reveal archeo-bacteria in the
interstices thereby providing insight into the development of photosynthetic bacteria.
Many suggestions for additional experiments follow
logically from our model for life's origin. We hoped our
readers would pursue them.
NORTH PACIFIC MINE PLUME
The year Akira was born, 1925, led to a peculiar effect
on his adult behavior. I first met Akira in person in 1972
over a large chicken dinner in our Dallas, Texas, apartment. My wife Susie, our daughters Wendy and Lisa
and I ate modest portions of the chicken and rice entree
while watching in respectful but disguised fascination
18
while Akira finished his large servings.
As Akira's scientific reputation grew, stories of his
appetite became legendary, yet all the while he
remained rail thin. Only after Akira died did I understand how his curiosity about drifting mines was connected to this interesting gastronomic behavior. Akira,
born and raised in Tokyo, turned 20 years of age when
General Douglas MacArthur arrived aboard the USS
Missouri to sign the treaty ending WW II hostilities. In a
few more weeks or months the Japanese population
might have faced widespread starvation, in part
because the Allies' Operation Starvation had paralyzed
Japanese shipping by air-deploying 12,026 submarine
mines during April-August 1945 (Sallagar, 1974). Had
the war continued another year it was estimated that
7,000,000 Japanese, Akira possibly among them, would
have starved to death (Johnson and Katcher, 1947).
To counter the communists' promises to provide food
in return for votes, General MacArthur had shipped in
many boatloads of food. With each ship's arrival Akira
related to me how he ate voraciously, usually in the
evening. This initiated the habit I observed decades
later.
Though the surrender on September 2nd, 1945 officially ended hostilities with Japan, damage to the large
island continued during September and October from
severe typhoons which also delayed extensive mine
sweeping operations until December (Dimitrijevic and
Ingrain, c. 1945). Overall, although the number of
cyclones for 1945 (35) was somewhat above normal for
the western North Pacific area (31.4), the typhoons
tracked northward over Japan more frequently.
Because these severe typhoons came at a critical juncture in WW II, the U.S. Navy issued a report prefaced
by: " . . . an enormous amount of material damage was
caused by several of these storms; and, had not Japan surrendered when she did, these typhoons could well have had farreaching and adverse effects on the course of the war
(Dimitrijevic and Ingrain, c. 1945)."
Typhoon damage beneath the sea surface went mostly unobserved. High winds and waves tore free thousands of mines from their moorings, setting them bobbing with deviMike protruding horns in the ocean
waves on the Kuroshio Current. As the typhoon season
ebbed, mine sweepers removed 4,085 mines but discovered that most of the deadly steel spheres could not be
located. Of the number of mines planted (15,130), 11,045
were left unaccounted for in seven harbor areas along
the southern and eastern shores of Japan. Thus, only
27% of the mines planted were deactivated (based on
computations for areas 3, 4, 5, 7, 9, 15, 23 in Lott, 1959).
All totaled, during 1943-1945 the Allies deployed
26,000 offensive mines around Japan and occupied territories, and Japan seeded an additional 51,000 mines
(Johnson and Katcher, 1947; Hartmann and Truver,
1991). In 1948, the U.S. Navy officially estimated that
35,000 active mines were adrift on the North Pacific
Ocean (Bristol, 1948).
Oceanography • VoL 12 • No. 1/1999
Akira sometimes
reflected as to what
the fate of these
mines might have
been. Subsequently,
we attempted roodeling the advection
and diffusion of the
mines across the
North Pacific Ocean.
After Akira died, W.
James Ingraham., Jr.
of theNOAAAlaska
Fisheries
Science
Center programmed
the OSCURS (Ocean
50"
40°
120° W
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,
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•
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140"
160" 180° 160" 140°
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j
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Pacific gyre; or (3) in
the Gulf of Alaska
gyre
along
the
Alaska Peninsula.
OSCuRsEXtendedruns
showed that after five
years the currents
spread them over
the North ,Pacific'
some ha, ing returned to Asia. To quantify the sightings of
mines at sea, the
mean and standard
deviation of the
sightings' latitudes
within bands of lon-
Surface
CUR
rent
/
/
/
10°
Simulations)
model
/
'"
/
I
to reconstruct the
gitude were comdrift of the floating
puted from marN TH PACIFIC/-OCgAtW--- /
"
mines.
iners' reports to the
He selected 18
U.S. Navy, Hydropoints at 60-mile Figure 8. Trajectories of mines set adrift along Japan bey.imzing 1 August 1945 according to
graphic Office durintervals along the the OSCURS model (by W. James lngraham, lr.). Dr!tiers were released at 18 locations alon~
ing February 13th
Japan (dots) and allowed to dr!ft.&r three years. Symbols (circles, diamonds, triangles)
Pacific Coast of coastal
through
August
indicate the mines'positions on the amm,ersary dates of the release.
Japan at the begin31st 1946, as shown
ning (1 August
on North Pacific
1945) of the unusual typhoon season toward the end of
Pilot charts (U.S. Navy Hydrographic Office, 1946;
Operation Starvation (Figure 8). Using OSCURS, the
Figure 9).
daily trajectories of drifters drogued at 20 m were simuDuring this 6-month interval, the plume's center
lated for three years using daily surface current fields
latitude and north-south width (two standard deviacomputed with empirical functions. These were calcutions) both increased with distance eastward across the
lated from monthly (1 August -31 December 1945) and
North Pacific Ocean. Because the 506 mines observed
daily (1 January 1946-1 August 1948) gridded sea level
represented only 1.4% of those estimated to be adrift
atmospheric pressure data and long-term mean
(35,000), we were puzzled as to the whereabouts of the
geostrophic currents (for OSCURS' description, see
98.6"/o of the mines not reported.
Ingraham
and
Miyahara,
1989;
see
also
The probability P of sighting at least one mine from a
http: / / www.refm.noaa.gov / drafts / oscurs / default.htm).
vessel passing east-west through the mine plume is
These simulations assume zero windage because
P~- 1- [1-(2w/4s)] %
(l)
mines drift mostly submerged as a result of their high
bulk specific gravity (note the water line evident on the
where w is the distance at which sailors routinely
beached mine shown in Figure 7). The OSCURS simulaspotted mines, s is the standard deviation of the mines
tion illustrated that mines set adrift during August 1945
in the north-south direction averaged over the North
could have reached the coastal waters of North America
Pacific (i.e., the average of the dashed line in Figure 9),
two to three years later (Figure 8). The number of mines
and n is the total number of mines adrift (derived from
reported washing ashore on Oregon and Washington
Hartmann and Truver, 1991, p. 193).
beaches was roughly proportional to the cumulative
In this formulation the 4s width of the plume
number of offensive mines laid by the Allies two years
embraces 95% of the mines, assuming a normal disearlier: Nine washed ashore during October-May 1945tribution in the latitudinal direction. Substituting w
46 out of 2,000 mines planted by August 1943; 5 beached
= 100 meters, s = 481 kilometers (from Figure 9), and
during 1946-47 from all 6,000 mines released up to
n = 35,000 mines, yields P = 0.974. In other words,
August 1944; and 63 reported aground during 1947-48
observers aboard each vessel had approximately a
of the total 25,000 planted by August 1945.
97% chance of spotting at least one mine during a
Model simulations indicated that those mines which
single transpacific crossing. It appears that the numdid not come ashore in North America circled back
ber of mines reported was limited by the n u m b e r of
toward Asia in three branches: (1) by recirculating immevessels observing the plume. Specifically; the 5{)6
diately south of the Kuroshio extension; (2) along the
mines s h o w n on the Pilot Charts were probably,
California coast, then toward Hawaii in the great North
r e p o r t e d bv a p p r o x i m a t e l y 522 vessels. In
Oceanography • Vol. 12 • No. 1/1999
19
the plume expansion is not linear, but varies with the
the first three years after WW II, 251 ships from many
winds and currents. In the future, Jim Ingraham and I
nations struck mines in the Atlantic and Pacific Oceans,
will continue exploring the thoughts of Akira.
and 116 either sank or were total wrecks (Lott, 1959).
By numerical experiFor comparison, by
I
i
.
.
ments we plan comparredefining 2w as a vesisons of the mine plume
sel's beam width, equaMINE PLUME C H A R A C T E R I S T I C S
and additional OSCURS
tion (1) provides the
simulations to explain
probability that a ship
S
the obvious differences
might actually hit a
~
between the present
mine. Assuming 2w=10
simulation and the stameters, yields 17% as
MEAN
r--tistical characteristics of
the probability that a
vessel would contact a
4o
~
the observed mines. We
1-38 r . . . .
also hope to locate addimine during a transpaz
26
28
3oo
tional mine sightings as
cific crossing within the
shown on Pilot Charts
mine plume. Assuming
~""~
NORTH - SOUTH PLUME WIDTH
(STANDARD DEVIATION)
issued after August
further that the rate of
1946, but which thus far
522 vessels crossing the
160°W
I
'
120°E
14'o°
160°E
180 °
,,0o
~20o2 5w°
have not been located.
Pacific per 6.5 months
LONGITUDE
We hope colleagues
applies to the three
Figure 9. Mean (solid) and standard deviation (dashed) of 506 mines sighted during
knowing the whereyears following WW II,
13 February - 31 August 1946, within selected bands of longitude. The east-west
yields 2,891
vessel
abouts of these charts
width of the bands was varied in order to obtain a sample size exceeding 25 (correcrossings, 491 of which
will contact us.
sponding mmlber of observations given along tlre dashed line). Tlre mine statistics
potentially could have
Picasso once said his
were computedfrom sightings shown on the North Pacific Pilot chartsfor the period
struck mines. The numgreatest ambition was to
13 February throuyh 31 August 1946.
ber of vessels actually
view the world through
striking mines was substantially smaller because crews
the eyes of a child. Akira came the closest of any person
maintained alert mine watches and the U.S. Navy
I knew as evidenced by his continual association with
swept the North Pacific shipping lanes.
young students from many disciplines, and his great
Despite great efforts, hundreds of mines went undecuriosity about subjects as diverse as the origin of life,
tected and floated long enough to wash ashore on
the history of WWII and his voracious dinner appetite.
North American beaches (Figure 7). During 1955 and
1956, eight mines floated ashore in Hawaii, some of the
A CKNO WLED GMENTS
last live WW II mines left afloat (Lott, 1959). By linearly
I thank Vincent J. Cardone, Oceanweather Inc., Cos
proportioning the Oregon-Washington total of 63
Cob, Connecticut, USA for information regarding
during 1947-48, to 1 in 1955-56, I obtained a total of 267
typhoons affecting Japan during August-October 1945
mines washed ashore in Oregon and Washington durand Mike O'Connell for providing the pumice shown in
ing the decade following WW 1I. Applying Hata's
Figure 6.
(1963) result for drift bottles released in the vicinity of
the mined areas with 44% found in British Columbia
and Alaska and 56% in Oregon and Washington, yields
REFERENCES:
a total of 477 mines estimated to have washed ashore in
Bristol, J.A., 1948: Here come the Jap mines. The Saturday
North America, equivalent to 1.4°/,, of the total 35,000
Evening Post, 20 March 1948, Vol. 220, 12.
mines thought to have originated around Japan.
Dimitrijevic, W.J., and J.R. Ingram, circa 1945: A study of the
typhoons of tire western Pacific, August to October, I945. Report
CLOSING THOUGHTS
prepared for the U.S. Navy (copy supplied by Vincent J.
Cardone).
Because Akira studied drifting particles much of his
Ebbesmeyer, C.C, A. Okubo and J.M. Helseth, 1980:
life, it is fitting to close with an overview of the mine
Description of iceberg probability between Baffin Bay and
the Grand Banks using a stochastic model. Deep-Sea
plume. As the mines drifted, they traced out a plume
Research, Vol. 27, No. 12A, 975-986.
according to the mixing processes operating across the
Ebbesmeyer, C.C., W.J. Ingraham, Jr., R. McKinnon, A. Okubo,
North Pacific during 1943-1946. From off Japan to the
R. Strickland, D-P. Wang and P. Willing, 1993: Bottle appeal
proximity of North America, the plume width increased
drifts across the Pacific. EOS, transactions of the American
from approximately 322 to 666 km (Figure 9).
Geophysical Union, Vol. 74, No. 16.
Hartmann, G.K. and S.C. Truver, 1991: Weapons That Wait:
According to the OSCURS model, individual mines
Mine Warf.ere in the U.S. Navy. Naval Institute Press,
could have drifted across the North Pacific for two to
Annapolis, Maryland, 345 pp.
three years. Assuming that the mines lay within 2 stanHata,
K., 1963: The report of drift bottles released in the North
dard deviations of the mean drift path, the plume
Pacific Ocean. The Journal of tire Oceanographical Society of
widened at the rate of 1.5 kilometers per day. Of course,
Japan, Vol. 19, No. 1, 6-15.
40 °
I-ATIIT~4I~
35
. _ _ J
--
3 5 ~ ua
43
r-- ~
t _ ~ 2
'
20
Oceanography • VoL 12
• No. 1/1999
Ingraham, W. J., Jr. and R. K. Miyahara, 1989: Tunin~ of
OSCURS nuaterical model to ocean surface current measuremerits in the Gulf of Alaska. U.S. DeE. Commer., NOAA Tech.
Memo. NMFS F/NWC-168, 67 pp.
Johnson, E.A., and D.A. Katcher, 1947: Mines against Japan.
Naval Ordnance Laboratory, White Oak, Silver Spring,
Maryland, 200 pp.
Lott, A.S., 1959: Most dangerous sea, a history or mine wa;?~re, and
an account of U.S. Navy mine warfare operations in World War II
and Korea. U.S. Naval Institute, Annapolis, Maryland, 322 pp.
Okubo, A., C. Chiang and C.C. Ebbesmeyer, 1977:
Acceleration field of individual midges, Anarete pritchardi
Kim within a swarm. Journal of Canadian Entomology, Vol.
109, pp. 149-156.
Sallagar, EM., 1974: Lessons fro;;; an aerial mining campai,?n
(Operation "Starvation'). U.S. Air Force Project Rand Report
Number R-1322-PR, 80 pp.
U.S. Hydrographic Office, 1946: Pilot charts of the North
Pacific Ocean, Number 1401. Issues for June through
November 1946 report mine sightings for 13 February
through 31 August 1946.
take his course. Smiling, I explained my situation and
respectfully declined. Bowing again, he ever so politely
offered to allow me to audit the course, but implored
me to register for it. I simply could not refuse this man,
w h o o v e r f l o w e d with such kindness, respect, and
humility. Needless to say, I submitted.
This turned out to be a very good decision, for that
course changed the way I think about particle dispersal
and transport processes in the ocean. 1 do not consider
mvself to be a mathematician by any stretch of the
imagination. Professor Okubo, on the other hand, was a
superb physicist and mathematician. While taking that
course, however, I was able to u n d e r s t a n d with ease
e v e r y concept presented. Why? It was because of
Professor Okubo's approach to his subject, his style of
teaching, and the principles upon which he focused,
rather than the details.
He began the course by respectfully acknowledging
that the students were comprised of non-mathematicians and non-physicists, and that the course was interAkira Okubo - The Man Inside the Man
disciplinary. He firmly stated his belief that one should
not need to be a mathematician or physicist to underPaul W. Sammarco
stand the concepts he was about to teach. He went on to
Louisiana Universities Marine Consortium (LUMCON)
say that he was determined to teach the sophisticated
Chauvin, Louisiana, USA
material in this course in a m a n n e r so that each student
could u n d e r s t a n d each and every concept presented - as
INTRODUCTION
well as u n d e r s t a n d the essentials of the mathematics,
irrespective of his or her background.
Who was this man Akira Okubo? Who was this m a n
In that course, Akira taught each principle until he
whose w o r k on the role of diffusion in ecology and
was satisfied that each student in the class u n d e r s t o o d
o c e a n o g r a p h y influenced so m a n y researchers a r o u n d
it. The total a m o u n t of material covered in the course
the world? Who was this m a n w h o was able to relay
was less important to him than the complete undergreat w i s d o m and guidance with only a few words and
standing of what he taught to each student. I had never
the mere nod of a head? Who was the m a n inside the
experienced this before in m y previous ten years as a
man? The p u r p o s e of this tribute to Professor O k u b o is
university student. He had the w o n d e r f u l ability to prenot to answer these questions, for I don't think a n y o n e
sent complex mathematical concepts in the most easily
can do that but Akira himself, but to provide insight
assimilable fashion. For someone whose first language
into the type of person he was - a great person.
was not English, he was an amazingly gifted and
effective speaker.
REFLECTIONS
Another time, I traveled to N e w York from Australia
Embarrassingly enough, it was m y fifth year as a
and visited Akira. We discussed our respective research
Ph.D. student at the State University of
projects at the time, particularly one
N e w York at Stony Brook. I was strugof his regarding the relationship
W h o was this m a n w h o s e w o r k
gling to keep u p with assisting in two or o;i the role o f diffusion in ecology b e t w e e n particle size, shape,
three courses, while analyzing data and
R e y n o l d s n u m b e r and sinking
and oceanography influenced
writing m y thesis. Time was scarce, and
velocity in the ocean. The results of
so m a n y researchers
the last thing I n e e d e d was additional
the study had astounding implicacourses. M y s u p e r v i s o r , Jeffrey S.
tions - and m a n y potential applicaa r o u n d the world?
Levinton, asked me if I w o u l d enroll in
tions. I then asked him about his
a new course being offered entitled "Diffusion in
research funding. Akira smiled and b o w e d and said he
Ecology," by a n e w faculty m e m b e r at the Marine
had received no funding for research that year. I told
Sciences Research Center - Akira Okubo. The a n s w e r
him I was sorry. He smiled again and said that there
was " T h a n k you, but no thank y o u . " The first request
was no need to be sorry; he picked u p the simplest of
was rapidly followed by a second, with a similar reply.
writing utensils which we h u m a n s have been using
Within a day or two, a short, demure, slight-of-build,
since the d a w n of western civilization - a pencil and a
gray haired, soft-spoken Japanese gentleman appeared
pad of paper - and staring me in the eye with that look
at m y office door. It was Professor Okubo. Bowing, he
of peace, calm, and wisdom - said - "This is all I need
introduced himself ever so politely and invited me to
to do my research. I do not need funding." This was a
Oceanogrophy • Vol. 12 • No. 1/1999
21
man so self-sufficient and humbly self-confident, that
rored by the way he led his life, despite its underlying
nothing could come between him and his science.
complexities. I learned so much from him whether by
Perhaps the one time I had the
climbing a mountain together or chatI sat and watched a nzan
honor of getting to know Professor
ting over a cup of tea by the fireside.
ahnost twice m y age
Okubo the best was when he accepted
He was a noble person who based his
my invitation to be the keynote speaklife on simple principles, holding true
walk sprightly, cane in hand,
er at the Boden Conference in
to them. He lived in the most humble
at an accelerating pace
Australia (equivalent to the Gordon
of abodes - a one-room flat beneath a
up the highest mountain
Research Conference in the USA).
house - because he simply felt that
in Australia
John Andrews, Malcolm Heron and I
was all he needed. Life challenged
organized the meeting at a ski resort in
Akira continually, and he responded
until he disappeared
Thredbo, New South Wales. Akira graby meeting those challenges one after
into the mist.
ciously accepted our invitation and
another with great inner strength. This
made the long journey to Australia for the meeting of 23
was particularly evident in his last few years w h e n he
invited researchers.
fell victim to cancer and exhibited a willingness to try
The topic of the conference was the bio-physics of
any and all options available - no matter how risky or
marine larval dispersal. The meeting was a gathering of
untried - to thwart this malady.
biologists and physicists working in teams on a variety
of larval dispersal problems. An interdisciplinary
CLOSING COMMENTS
approach to solving such problems was emphasized, an
Akira Okubo was an amazing man and he had a proapproach professed by Dr. Okubo for years. Five differfound impact on my life - both personally and profesent phyla were examined, and two research teams were
sionally. He leaves behind an enduring legacy in his
represented within each group. The meeting design, I
applications of diffusion theory to the fields of ecology
thought, was ideal. What I didn't realize at the time was
and oceanography. These concepts he gave to science
that Professor Okubo had planted the seeds for such an
unselfishly, apparently risking his career by entering
approach to scientific problem solving in my mind some
the field of ecology as a physicist, after completing a
15 years earlier. He had a unobtrusive way of sowing
sabbatical to conduct research on what he considered a
ideas in one's subconscience, which might emerge years
very exciting new interdisciplinary field - the applicalater in some project. The presentation of his keynote
tion of diffusion physics and ecology to the study of
paper was, as usual, eloquent. It was subsequently pubinsect swarming.
lished (Okubo, 1994) in the book based on the conferI miss Akira as a friend and a colleague. I am so
ence proceedings (Sammarco and Heron, 1994).
thankful, however, for the good fortune of having
Akira was a m a n with purpose, determination, and
known him and worked with him. I feel certain that,
great inner strength. The meeting was held in February,
somewhere up there, he is walking with a cane in his
during the austral summer. Mt. Kosciosko, the highest
right hand and a pencil and pad in his left - smiling and
mountain in Australia, was almost devoid of snow. We
saying, "This is really all I need."
had reserved two half-days for the participants to relax,
hike, experience the beauty of the local countryside and
REFERENCES
establish friendships with each other. Dr. Okubo, then
Sammarco, EW. and M.L. Heron (eds.), 1994: The Bio-Physics
in his mid-60s, decided to hike to the peak of Mt.
of Marine Larval Dispersal. Coastal and Estuarine Studies,
Kosciosko. I offered to join him and he agreed. He wore
American Geophysical Union, Washington, DC.
a small green peaked felt hat, a flannel shirt, some hikOkubo, A., 1994: The role qf d!ffusion and relatedphysical processes
ing pants, boots, and a cane. He started at a fair pace
in dispersaland recruitment qfimarine populations, ibid., pp. 5-32.
and held that pace, walking continuously uphill. I kept
up with him until about one half to two thirds of the
way to the top of the mountain. I requested a rest. He
Reminiscences
of a Former Student
agreed and waited. We started again.
Shortly after that, I needed another rest. I realized I
Brian G. Sanderson, School of Mathematics
was only holding him back and encouraged him to go
University of New South Wales
ahead without me, as I would only be a hindrance. He
Sydney, New South Wales, Australia
smiled and agreed to leave me there while he continued
up the mountain. And so, with mouth agape, I sat and
My association with Akira began in August 1979
watched a man almost twice my age walk sprightly,
w h e n I undertook graduate studies at the Marine
cane in hand, at an accelerating pace up the highest
Sciences Research Center (MSRC) at Stony Brook. I had
mountain in Australia until he disappeared into the
read some of Akira's publications on differential kinemist. I expect that this is how he entered heaven.
matic properties of surface currents and their effect on
A few more comments. The simplicity with which
eddy-diffusion in the ocean. These works had been useAkira presented complex scientific concepts was mirful for interpreting drifter tracks I had measured in
22
Oceanography • VoL 12
• No. 1/1999
New Zealand during my Master's degree research at
the University of Auckland.
Akira was looking for someone to help with drifter
studies for the so-called Lagrangian Eulerian Diffusion
Study (LEDS) funded by the US Department of Energy.
Malcolm Bowman, whom I had met on his sabbatical
visit to New Zealand that year, invited me for Ph.D.
studies at MSRC. He took care of all the formalities, put
me in touch with Akira and within a month I found
myself at Stony Brook.
I did not initially appreciate how accessible Akira
was. As a beginner I was overwhelmed by Akira the
renowned theoretician. But Akira practiced a very compassionate, very human, type of science. Working with
Akira just ended up being such a natural thing to do.
Oceanography at MSRC was very multidisciplinary
and it seemed that Akira could construct an elegant,
fundamental mathematical theory to describe almost
anything he was challenged with. Sometimes his theories were a little unconventional, even titillating, as in
his manuscript: Sperm Flux and Dispersal: A Diffusion
Model (which was never published for some reason).
The point is that Akira saw the little twists and turns
that make science fun; he always had a witty and imaginative perspective. Many at MSRC would tread the
path to Akira's office, finding him hiding amongst his
many books and papers.
Figure 10. Master and disciple. Akira and Brian
Sanderson hiking over the hills of Newfoundland, Canada,
ca. 1986 (photoby DeborahBray).
One day I presented Akira with several pages of
mathematical notes that turned out to be an erroneous
proof of some now forgotten issue to do with flow field
singularities. Akira used this as an opportunity to show
me how some of my mathematics was related to a
Hamiltonian formulation of a totally different problem;
my error was exposed as an exciting process of discovery! From that day on I regarded Akira as my intellectual father.
Science could be described as a conservative and
highly critical method of formulating and testing ideas
that seem to work in explaining the mysteries of nature.
It is fortunate that the scientific profession is occasionally graced by gentle people like Akira. There is someOceanography • Vol. 12 • No. 1/1999
thing noble about a person who can undertake the
critique, produce the ideas, and in the process make
everyone associated with them feel good and that they
contributed in a meaningful way to the study.
Thank you Akira, you made science joyful.
Akira's Visit to St. John's,
Newfoundland
Ian Webster
CSIRO Land & Water
Canberra, Australia
During 1986 while I was a faculty member in the
Department of Physics at Memorial University in
Newfoundland, Canada, Akira paid the department a
visit and presented an afternoon seminar there. After
the seminar it was decided to take Akira out for dinner
to a pizza restaurant, downtown, some distance from
the university. The only problem was that of the nine of
us who wished to attend I was the only one who had a
car, and it was a small Japanese hatchback at that.
There was nothing to do but for all of us to squeeze
into every nook and cranny of the car. Brian Sanderson
and one of our graduate students being the largest
members of our group had the luxury of occupying the
trunk which left seven for the car's interior. I had a two
year old daughter at the time, whose baby seat occupied some of the back seat. In order to squeeze everyone
into the car it was necessary that the baby seat be occupied. Only Akira was slim enough to fit into it, and sit
in it he did. Off we went.
It was a measure of the humility and serenity of Akira
Okubo that he was totally unfazed by this bizarre and
dangerous situation. To this day I have an image in my
mind of an eminent Japanese gentleman dressed in
jacket and tie sitting in the baby seat as if it was an
entirely normal thing to do.
N o t e s o n S i m u l a t i o n of D i s p e r s i o n
in an E d d y Field
C o v e r i n g a W i d e Spectral R a n g e
Gerrit C. v a n D a m
National Institute for Coastal and Marine
Management/RIKZ
Den Haag, Netherlands
Akira Okubo must have been fascinated by the kinematics of dispersion. The number of reports and articles
on the subject published under his name, quite often as
the only author, is very large. He rightly paid much
attention to the phenomenon of dispersion by the
combined action of a shearing velocity and a mixing
mechanism driven by velocity variations on a small
23
scale, such that the latter can be treated as a "classical
(Fickian)" diffusion process.
On the one hand Akira was rather fond of finding
analytical solutions for well-defined but of course simplified cases; on the other hand he was well aware of
the complex conditions in physical reality. This is
reflected not only in his gathering and ordering of large
sets of experimental data but also in formulating the
restrictions for a hypothetical case, restrictions to make
it analvticalh, tractable.
Figure 11. Akira Okubo sitting on the first row (next to J. T. F.
Zimmerman) at the international workshop on spectral characteristics
of velocity variations in the sea and their significance for dispersion and
mixing (The Hague, 1994). Behind him in the second row is his great
Russian counterpart, the late Rostislav Ozmidov (next to Konstantin
Korotenko).
As an example, in his paper (Okubo, 1966) dealing
with an instantaneous source in a two dimensional
velocity field with uniform shear combined with classical isotropic diffusion, Akira relates these conditions to
physical reality as follows:
To construct a working model of diffusion, we propose a hypothetical spectrum of turbulence which
consists of two major parts: the large scale eddies
and the small scale eddies. The two eddies are sufficiently separated in scale so that the scale of
diffusion lies somewhere between the two scales
of the eddies.
For the analytical solution to be worked out in the
paper, the assumption is m a d e that the 'small scale' is
effectively of negligible dimension and the 'large scale'
is in fact one e d d y of infinite size (to obtain a uniform
shear). Akira was well aware that in the physical reality
one has a complete spectrum of e d d y scales. The word
e d d y as well as the idea of a spectrum have no relevance
to the rest of the paper if it is seen as a purely mathematical exercise. The introductory paragraphs clearly
show that, in this context, Okubo was thinking much
more as an oceanographer than as a mathematician.
The realistic case of simultaneous eddies of scales
from small to large compared to the scale of the phen o m e n o n to be studied, was u n d o u b t e d l y often on his
mind and he probably regretted that it seemed impossible to attack it analytically in a rigorous way. In a later
24
paper (Okubo, 1968) he transfers, in a heuristic way, his
exact solution for "the simple shear model" to the case
of a growing patch in a field with a "continuous spectrum of turbulence."
Deriving in this way some classical cases of time
behavior of patch size, Akira describes this result as an
illustration of the p o w e r of the "shear-diffusion model";
it seems to explain everything. The paper ends with
"Shear, shear everywhere!" (also see "Akira Okubo and
Shear Dispersion" in this issue, by Konstantin
Korotenko). But apparently he was still dreaming of a
more explicit and more powerful concept for the case of
a continuous spectrum.
When I visited Stony Brook in 1988 and presented the
results of a computer program in which complete spectra were explicitly simulated by series of harmonic
functions in space (generating a "synthetic e d d y field"),
he said that I had realized a dream of "every oceanographer." From this overstatement we might conclude
that it was at least one of his o w n dreams. The basic idea
of the model presented in 1988 in Stony Brook (Van
Dam, 1988) and some early results are given in Van
Dam (1980a); a first presentation took place in 1980 (Van
Dam, 1980b). For an explicit description including
equations refer to Van Dam et al. (1999).
In 1994 1 organized an international workshop in The
H a g u e on "Spectral characteristics of velocity variations
in the sea and their significance for dispersion and mixing" and was very h a p p y that Akira, on his way back
from Italy was able and willing to present the opening
lecture ("The historical view of oceanic diffusion,"
Okubo 1994) in which of course he gave due attention
to the synthetic e d d y concept. Okubo's contribution
was followed by "Oceanic Spectra," presented by his
great Russian counterpart Rostislav Ozmidov, with
w h o m Okubo published an article on the subject during
the time there still was an iron curtain between them
(Okubo and Ozmidov, 1970). In Figure 11 the two are
seen, sitting in the first and second row. This is an historical picture; a few weeks later Akira's serious illness
was discovered; O z m i d o v died in February 1998. I had
the luck to see Akira one more time, at the Okubo
S y m p o s i u m at Stony Brook in 1995 (Figure 12).
Figun' 12. Akira Okubo, DoJ/ l~rilch~zrdfh'flt) alld Gerrit van D¢ml,
photographed at the Okubo Symposium at Stony Brook in July 1995.
Oceanography • Vol. 12 • No. 1/1999
REFERENCES
Okubo, A., 1966: A note on horizontal diffusion from an
instantaneous source in a nonuniform flow. J. Occauo~r.
Soc., Japan, 22, 35.
Okubo, A., 1968: Some remarks on the importance of the
"shear effect" on horizontal diffusion. I. Occanogr. Soc.,
Japan, 24, 60.
Okubo, A., 1994: The historical view of oceanic diffusion:
From radially-symmetric to chaos-induced diffusion.
International workshop on spectral characteristics of velocitv variations in the sea and their significance for dispersion
and mixing, The Hague 8-9 March 1994 (complete set of
copies of Okubo's transparencies available on request).
Okubo, A. and R.V. Ozmidov, 1970: Empirical dependence of
the coefficient of horizontal turbulent diffusion in the ocean
on the scale of the phenomenon in question. Oceanology, 6,
308-309.
Van Dam, G.C., 1980a: Scale dependent dispersion of distinct
particles in an artificial eddy field. Rept. 07 80-FA, Physics
Division, Rijkswaterstaat, The Netherlands (colloquium
NIOZ, Texel, The Netherlands, April 17-18, 1980).
Van Dam, G.C., 1980b: Distinct-particle simulations. In
Pollutant Transfer and Transport in the Sea, Vol. I, G.
Kullenberg ed., CRC Press Inc., Boca Raton, Florida.
Van Dam, G.C., 1988: Seminar at MSRC, Stony Brook, April
12, 1988.
Van Dam, G.C., R.V. Ozmido~, K.A. Korotenko and J.M.
Suijlen, 1999: Spectral structure of horizontal water movement in shallow seas with special reference to the North
Sea as related to the dispersion of dissolved matter. J. Mar.
Syst., in press.
Random
Walks with a Hana-Like Bias
James G. Mitchell
Department of Biological Sciences, Flinders University
Adelaide, Australia
A m o n g his m a n y qualities, Akira was well k n o w n for
his bike riding, not o w n i n g a car and not driving. It will
come as a bit of a surprise to some then to find out that
Akira was an excellent controller of cars and that on the
occasional w e e k e n d e v e n i n g he w o u l d slip into
Manhattan, N e w York City, to help direct m i d t o w n
traffic.
Akira's automotive talents were revealed to me the
first time we visited Manhattan. We were on our way to
the American M u s e u m of Natural History. I had been at
Stony Brook for about 2 m o n t h s and Akira was not yet
my Ph.D. co-supervisor.
Apart from the trip on the Cross Bronx Expressway,
linking N e w Jersey to the bridges to Long Island, which
I had taken when first m o v i n g to Stony Brook, this was
the first time I had driven in N e w York City,. Being from
Los Angeles, m a n y long h o u r s spent creeping in
straight lines along California freeways had prepared
me for N e w York traffic, or so I thought.
Soon after arriving in m i d t o w n Manhattan. I was feeling a little bit daunted. Akira was sitting in the middle of
the backseat, muttering something about all the people
who lived in crowded New York City. But as always he
Oceonogrophy • Vol. 12 • No.
1/1999
was sensitive to those around him. Perceiving my tension, he poked his head forward as we entered Times
Square and said something to the effect of: "Don't worry;
just drive as if there are no other cars on the road." H o w
much our car, cutting across lanes, contributed to the
cacophony of horns in Times Square I do not remember.
Foreshadowing the letters I would find left on my
desk over the next few years, Akira, as we drove, refilled
the Manhattan-car model with comments and crucial
pieces of information. The thrill of driving in Manhattan
seemed in part to spring from the helter-skelter d y n a m ic of cars approaching very close together and then dispersing. It seems he relished investigating this kind of
point and counterpoint, giving it, for example, the wonderful label of "interplay" in his "Fantastic Voyage"
paper about microscale processes. He seemed truly,
delighted with that evening and our m a n y subsequent
trips to Manhattan, Cornell University in Ithaca and
most frequently, the restaurant Hana, located in Port
Jefferson, Long Island, not far from Stony Brook.
As a graduate student, the fantastic voyage for me
was working with Akira on the motility of marine bacteria in a turbulent ocean. His analysis went to the crux
of this interplay with characteristic sharp, u n a d o r n e d
identification and analysis of the vital principles in the
process. During m y early studies at Stony Brook I was
working in Jed Fuhrman's lab, trying to count the number of bacteria in microzones (bacterial swarms around
phytoplankton or particles). The results were always
equivocal at best and in part this was because it was
u n k n o w n exactly what microzones looked like.
A model was n e e d e d and Akira enthusiastically,
joined in, generously sharing his knowledge and skills
of quantitative examination. Much to m y surprise, it
soon became clear h o w similar the microzones problem
was to midge swarming [Alan Elliott; this volume] and
wildebeest herding [Simon Levin; this volume].
In 1984, H o w a r d Berg's Random Walks in Bioloy,y had
just been published. The quantitative portrayal of the
simplicity of bacterial motility fascinated Akira, as did
the potential for analytically modeling creatures that
lacked vision, feeding behavior, mating rituals and
predator avoidance. Tracing the line of research on environmental motility led us back to the parent concept for
bacterial swarming, the idea that soil bacteria congregate around legume roots, creating a "rhizosphere."
Here was sweet serendipity.
In the soil beneath his beloved m i d g e swarms,
bacteria teemed to form their own tiny swarms. This
realization was more than enough for a celebration at
Hana. We teased apart the evidence for microzones over
sashimi, adding back-of-the-napkin diffusion calculations for p h o t o s y n t h a t e excretion, p h y t o p l a n k t o n
movement, chemotactic drift and turbulence. As the
research progressed, trips to Hana mounted. From them
emerged the view of bacteria fighting against rapid spatial dispersal and s e p a r a t i o n in all but the most
quiescient parts of the ocean.
25
When our paper on microzones appeared in the journal Nature (Mitchell et al., 1985), there was no special
trip to Hana. The celebration and enjoyment had
already taken place. Akira was such an excellent navigator in traffic jams and so enjoyed r a n d o m walks
because it was the voyage and not the destination that
provided him the most exhilaration.
REFERENCES
Berg, H.C., 1983: Random Walks in Biology. Princeton
University Press, Princeton.
Mitchell, J.G., A. Okubo and J.A. Fuhrman, 1985: Microzones
surrounding phytoplankton form the basis for a stratified
marine microbial ecosystem. Nature, 316 (6023), 58-59.
Okubo Sensei
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Figure 13. Akira's sketch map with travel instructions to Haha (Hana)
restaurant in Pot Jeanson (Port J~fferson), Long Island, New York.
We did not make it to the American M u s e u m that
night, or ever for that matter. Despite enjoying m a n y
r a n d o m walks together, thoroughly exploring the city's
local environment, that particular location eluded us.
Like marine bacteria, however, there was a bias to our
walks, and as with bacteria the bias was towards a
fundamental energy source, food! sashimi! This was
fortunate for while we celebrated our bacterial discoveries, we always agreed that none of the distant
microzones had the quality, atmosphere or memories of
Hana (Figure 13).
26
Akira had the corner office in E n d e a v o u r Hall at the
Marine Sciences Research Center (MSRC) at Stony
Brook. You w o u l d always see him working there late
each night, s u r r o u n d e d by his books and papers. The
F u h r m a n laboratory was also in E n d e a v o u r Hall. I am
quite a night owl myself, so Akira and I quickly developed a routine where we either talked in his office in the
evenings or went for a Japanese meal at the H a n a
Restaurant in nearby Port Jefferson. During m y Ph.D,
studies with Jed Fuhrman, that routine p r o v e d to be
very important to me, both personally and professionally, in the m o n t h s and years that followed.
Shortly after I started at MSRC, m y father s u d d e n l y
passed away. I n e e d e d some time at h o m e with m y
mother but after three months, I started to feel that this
kind of time wasn't good for me. I came back to MSRC
the following spring but was probably ill-equipped
psychologically and emotionally to pick up where I had
left off. However, I knew I needed some way to focus
m y thoughts, something I could concentrate on.
That's w h e n Akira began (what appeared to be at the
time) these pointless conversations with me. He w o u l d
drop by my office in the evenings and start talking
about the tree outside his office, some childhood memory or eating Maryland blue crab. He never asked about
m y father. He simply kept u p these one-sided conversations. I w o u l d sit there, not really taking anything in,
but w o n d e r i n g h o w in the world this busy fellow had
so much time to chat. Of course, over the next few
months, I began to laugh at his stories and feel myself
surfacing out of the depths of grief I had been in. Only
years later did I recognize h o w Akira, in his o w n way,
had been trying to help rehabilitate me that year.
Akira Okubo made other contributions in my life and
these were as a teacher and a scientist. In my doctoral
research, I studied the role of viruses in controlling
bacteria in the ocean. When Jed first suggested this project,
I could not imagine how I would relate the abundance of
marine viruses to an estimate of how many marine bacteria are killed by viruses. Yes, free viruses were clearly
abundant, I was getting a million to ten million viruses in
a milliliter of Long Island Sound seawater. But how to estimate virus-induced death in bacteria?
I took John Sieburth's lead and began to thin-section
Oceanography • Vol. 12 • No. 1/1999
marine bacteria and look at them under the electron
His questions finally registered and I realized that,
microscope, the idea being that I would have a "snapunlike living organisms, most of the life cycle of the
shot" of marine bacteria with viruses.
virus is invisible. It is only when the
These sessions
As many readers may know, electron
virus is ready to be released that it
became important lessons to me
microscopy is very tedious so I was
becomes visible. What I was seeing in
eager to show off any new micrographs
my micrographs was only the visible
because Akira forced me
I managed to get. I would run down to
part
of the virus cycle. This meant
to hone m y ideas in order
Akira's office and show him my latest
that many more bacteria than were
to explain them to him.
results. Of course, he could not make
visible to the eye (and to the electron
heads nor tails of the electron micrographs, and so began
microscope) were potentially infected with viruses.
my attempts at teaching Akira microbiology.
Akira immediately understood the implications of
this point and helped me develop a simple model for
estimating the actual number of bacteria in the sea
which are infected with viruses. Later, with data from
laboratory experiments conducted at the University of
Southern California, we found that something like 1015% of the life cycle of a marine virus is visible by
microscopy, which meant that on the order of tens of
percents of bacteria in the ocean are infected by viruses
and that viruses represent a major impact on bacteria in
the ocean (Proctor et al., 1993).
Akira's child-like curiosity and interest in all things
biological played a major role in my scientific training.
I still wonder today whether had he not quietly insisted
on those explanations, would I have recognized what I
Figure14. LitaProctor,Akiraand caninecompanionrestingon
was not seeing in the microscope? I feel lucky to have
the PacificCrestTrailin the SierraMountains, Cahi~rnia,1990
had Akira as a friend, grateful to have had him as a
(photo by Val Gerard).
teacher and privileged to have had him as a colleague.
As I explained microbiology to Akira, he would
frequently ask me, "Is that so? And why is that important?" Each time he asked this question, I would refine
the point I was trying to make, in order to reduce it to its
most basic elements. I felt at that stage of my studies that
my understanding of microbiology was weak; sometimes I felt I had explained well enough for him to
understand and sometimes I didn't. These sessions
became important lessons to me because Akira forced
me to hone my ideas in order to explain them to him. In
fact, Akira took enough interest in my work early on that
he and I together wrote a small paper on the dynamics
of virus/host encounters (Okubo and Proctor, 1989).
In the early stages of my research, I was initially
disappointed at one apparent result that was beginning
to emerge from my microscope work. A very low percentage - on the order of 1-5% -- of the bacteria in my
samples had viruses. With papers appearing in the
literature that protozoa were eating nearly all of the bacterial production in the water column, I thought this
probably meant that viruses were not very important in
the food web of the ocean.
This is where Akira's insights again proved crucial.
As with many earlier conversations with him about
microbiology, I would use diagrams. One day, I started
to draw the classical "life cycle" of a virus when I
suddenly realized why the percentage of infected bacteria in my samples was so low! He had often asked me
how this or that part of the virus cycle was different.
Oceonogrophy • VoL 12 • No. 1/1999
REFERENCES
Okubo, A., and L.M. Proctor, 1989: Marine bacteria and bacteriophage population dynamics. Aquabiology (in Japanese),
66, 17-19.
Proctor, L.M., A. Okubo and J.A. Fuhrman, 1993: Calibrating
estimates of phage-induced mortality in marine bacteria:
ultrastructural studies from one-step growth experiments.
Microb. Ecol., 25, 161-182.
Reflections
Stella Humphries
CSIRO Corporate Human Resources
Dickson, Australia
Akira Okubo was my external Ph.D. examiner. My
thesis, submitted in 1983, was on the quantitative relationships between turbulence, stratification and primary productivity of algal populations. As an ecologist
delving into the field of fluid mechanics, I remember
clearly the trepidation I felt when my supervisor (Jorg
Imberger at the University of Western Australia)
suggested Akira as an examiner! To me he was one of
the great masters in building bridges between ecology
and physics, with clarity that only profound insight and
wisdom brings.
But I am now glad that Jorg encouraged me to go to
the top, for my next impression was one of a man who
27
anecdotes from thirty years of living in America. His
took a deep personal interest in people. His w o r d s as
stories incurably infected me with his love for New
examiner were inspiring and uplifting and conseYork City.
quently gave me confidence to venTo me he was one of the great
At MSRC, I continued to anature forth professionally. I had the
lyze
three-dimensional
data of mosgreat privilege and joy of meeting
nlasters in building bridges
quito swarming. From these analyses,
Akira on a short visit to Stony Brook
between ecology and physics,
Akira and I started to construct a
in 1993. I was left with the impreswith clarity that only profound m a t h e m a t i c a l m o d e l for m o s q u i t o
sion that here was a man of great
insight and wisdom brings.
swarming.
kindness, humility and cheerfulness
H o w e v e r Akira left his earthly
he was an inspiration and a master
h o m e forever before we finished it. I still have m y notes
in life as well as in academia.
on the model; I hope to complete this project at some
point in the future.
Aside from research, Akira and I enjoyed chatting
Akira and Insect Aggregation
about m a n y things; insects, ecology and about his m a n y
adventures. Akira always loved to chat. He seemed to
become more eloquent w h e n talking in his mother
Terumi Ikawa
tongue, Japanese. Usually, good Japanese food accomDepartment of Liberal Arts and Sciences, Morioka College
panied these conversations. Akira would say, "I cannot
Takizawa-11111ra
live without rice! Having sashimi is even better!"
lwate-y, un, Japan
Looking at Akira eating and chatting pleasantly, I often
w o n d e r e d h o w he could be so productive both in mathIt was in the s u m m e r of 1990 w h e n I first met Akira.
ematical ecology and physical oceanography, fields that
During this visit to Japan, he visited the Tanashi
seemed so far apart!
Experimental Farm attached to the University of Tokvo.
One day, Akira s h o w e d me a paper about the oceanThere my collaborators and I were working on threeic insects Halobates. Like m a n y entomologists, I did not
dimensional m e a s u r e m e n t s of mosquito swarming.
know that the insects could exist in the open ocean.
Akira e n o r m o u s l y enjoyed the day he spent at the
Halobates are water striders which belong to the family
Experimental Farm. Just like a curious small child, he
Gerridae. The genus Hah)bates has 44 species and all of
was greatly excited, biking around the farm, munching
them live in marine environments. Five species of
on sandwiches, peering at our instruments for threeHalobates live in the open ocean. They are the only true
dimensional measurements and watching the mosquipelagic insects out of more than one million species of
toes swarming in the dark.
insects in the world.
Insect swarming (or more widely, animal aggregaI was so impressed with the distribution map of these
tion) was one of Akira's most favorite scientific topics.
five species of Halobates. It looked like a huge tropical
Akira loved to observe small insects swarming in the
belt girding the three major oceans. I have never seen
air. He was fascinated that the insects were never difsuch a vast distribution area without any discernible
fused but kept their swarming station, although they
break. It was almost impossible for me to believe that
appeared to be flying randomly. He often told me that
these insects could survive, find mates and reproduce on
he was so interested in insect swarming that he once
the ocean surface, with no shelter from wind and storm.
gave up his own career as a physical oceanographer so
I pestered Akira and the other oceanographers at
he could focus his attention on insect swarming. For
MSRC, asking m a n y questions about the properties of
some time he desperately tried to find an entomologist
the ocean surface and factors that could affect the surwho might collaborate with him. The fruits of his enthuvival of Halobates. Finally, I was successful in convincing
siasm and searching are his pioneering research on
Akira to take an interest in studying ocean striders. We
three-dimensional trajectories and behavior of midge
m a d e calculations about h o w far Halobates could drift
swarming conducted with the eminent entomologist
apart by oceanic diffusion and how often males and
H.C. Chiang.
females could meet on the ocean surface. We also calcuFrom 1991 to 1993, I had the pleasure of visiting the
lated the steady state condition among the population
Marine Sciences Research Center at Stony Brook. I am
growth rate of Halobates, the distribution size, and
always thankful that Akira and Jerry Schubel, the fordiffusivity. Based on the results of our calculation, we
mer Dean and Director of MSRC, gave me this great
surmised h o w oceanic diffusion could influence
opportunity; welcoming me, even though I am an entoHalobates' life history strategies (Ikawa et al., 1998).
mologist, not an oceanographer. It was m y first visit to
I wish I could sit and share with Akira m y recent
a foreign country. Akira gave me much helpful guidance about living in the United States; I know he did
work on Hah)bates and mosquitoes. I cannot, but still I
this for m a n y foreign visitors. He enjoyed telling me
am very grateful to have had the opportunity of workabout his life experiences, filled with m a n y interesting
ing with Akira on some of his favorite topics.
-
28
Oceanography • VoL 12 • No. 1/1999
REFERENCE
lkawa, T., A. Okubo, H. Okabe and L. Cheng, 1998: Oceanic
Diffusion and the Pelagic Insects Halobates spp. (Gerridae:
Hemiptera). Marine Biology, 131, 195-201.
Akira Okubo and Shear Dispersion
Konstantin Korotenko
P.P. Shirshov Institute ~ Oceanology
Russian Academy qfi Sciences
Moscow, Russia
Along with Ken Bowden of the University of
Liverpool, United Kingdom, Akira Okubo was a
pioneer in the development of ideas related to the
processes of horizontal shear dispersion in the sea.
Akira's mathematical skills led him to develop enlightening analytical representations for the concentration
fields for substances diffusing in shear flows. These
representations elucidated the dependence of the diffusion process on some basic flow parameters. Because of
their clarity they received wide acceptance and found
considerable practical use.
Akira's work influenced my own research too. I have,
for example, examined the horizontal spread of a
substance due to the interaction of an unsteady current
shear with transverse mixing in three spatial dimensions. My analyses have closely followed the derivations presented by Akira in the 1960s for a variety of
flows. Akira's basic development of the advective-diffusive equation provided a firm basis for further refinements. His decomposition of the local advective flow
velocity into a mean flow and a uniform shear (both of
which could be time varying) proved to be very fruitful.
His methodology also permitted the application of a
wide variety of initial and boundary conditions.
The solutions he obtained provided a mechanism for
understanding the nature of the interaction between current shear and transverse mixing in a particular flow.
His use of moments to interpret the behavior of the
analytical solutions and field data proved to be quite
powerful. More generally, his solutions provided a basis
for scaling and interpreting shear dispersion processes
for a wide variety of flows encountered in nature.
His works will endure, and I wish to thank Akira
Okubo in this tribute for the opportunity of meeting him
on both sides of the Atlantic Ocean and learning from
his great wisdom.
F r o m C o a s t to Coast: A k i r a O k u b o ' s
Trek A c r o s s A m e r i c a
Ft'otll Olle diltlt'llsioll" thc l l l a l l ,
to two dimensions: the path,
to three dimensions: the life well-lived.
Jeannette Yen
Marine Sciences Research Center
State University of New York
Stony Brook, New York, LISA
It is a fine summer's day in the month of June 1994.
Rudi Strickler and I wait at the station in Milwaukee,
Wisconsin. The whistle announces the approach of a
chugging train. Out pops our intrepid visitor: Akira
Okubo - with a beautifully tanned smiling face, khaki
shorts revealing strong legs with well-worn hiking
boots. No luggage except for his sturdy backpack and
First Recipientof okub0 Award Announced
Professor M a r t i n N o v a k of the Institute for A d v a n c e d S t u d y
at Princeton University, N e w Jersey, U S A has b e e n selected as the w i n n e r
of the first O k u b o A w a r d s p o n s o r e d b y the J a p a n e s e Association
for M a t h e m a t i c a l Biology a n d the Society for Mathematical Biology.
The O k u b o A w a r d , w h i c h is offered the first time this year, is to be a w a r d e d
e v e r y other y e a r to h o n o r alternately a d i s t i n g u i s h e d w o r k b y a scientist u n d e r 40
a n d lifetime achievement, w i t h the first a w a r d going to a scientist u n d e r 40.
The selection c o m m i t t e e cited Professor N o w a k for his p a p e r in Theoretical P o p u l a t i o n Biology
on stochastic strategies in the p r i s o n e r ' s d i l e m m a . In a d d i t i o n to its o w n merit
of i n t r o d u c i n g the effects of error or u n c e r t a i n t y in strategies,
this p a p e r triggered a stream of related w o r k on the e v o l u t i o n of cooperation.
Oceanogrophy • VoL 12 • No. 1/1999
29
his trusty umbrella. "This umbrella," he waggles at us,
"is so-o-o useful. Its shade keeps the hot sun from burning me. As a walking stick, it steadies m y descent d o w n
a craggy mountain path. When passing through a field
of tall grass, I can search out snakes ahead (he demonstrates by sinuously shaking the umbrella). And of
course, it keeps off the rain." He smiles that multifaceted smile. And we smile, witnessing once again the
multidimensionality that characterizes this great man.
Where was he going? Where did he come from? This
tribute describes only a fragment of a life well-lived. His
cross-continental walking journey provides only a few
clues, a skeletal framework. His intent was "to walk
across America." We couldn't imagine how this man who did not drive, who had ridden his bicycle to work at
the Marine Science Research Center - familiarly clad in
his big yellow goggles, helmet, windshell - through sun,
I
,° •
rain, sleet, and over and into potholes (once breaking his
nose!) - thought that he would accomplish this? Of
~-,,~'; If J
course, never in the conventional sense.
Akira would often travel on Thanksgiving, taking the
train to Huntington, a town west of Stony Brook on
Long Island, N e w York. He would then walk the 20
miles back h o m e to join his landlady's family for
,IS" -':,"
;
:
i
Thanksgiving dinner. That was one leg of his journey he
• q*
repeated every November. On other occasions, he
would visit his colleagues, including Simon Levin [this
~'~$
:volume], at Cornell University, in Ithaca, upstate N e w
York, then take off walking from there - in various directions. Over the years, Akira made more than forty visits
to his favorite American city: Seattle, where he wanted
to retire (if he ever would really retire from the fascinating career he led). There, in the Pacific Northwest of the
Figure I5. A page from Akira's sketchbook, illustrating various biologicalUnited States, Akira would hike in the rain forests,
physical interactions involving plankton. Clockwise from upper left: a
mountains, meadows, steep rivers and gorges.
copepod (N 1 ram) entrains water, signals and food within its feeding
cunent and releases particles and metabolites; a swarm of copepods can
Then there were those fortunate times w h e n he
occupy a space of 1 to 10 m; the movement of motile microplankton can
w o u l d come to visit wherever we lived or worked. If it
display gyrotaxis; a virus and bacterium rely on molecular d!ffusion;
were summer, we could inquire into his itinerary and
phytoplankton, ranging in size from 2 to 100 }un, also release various kinds
find several days with no activities planned. These were
of exudates into their surroundings.
the days he w o u l d escape from his four walls and
As our eyes adapted to the darkness, he stepped carecontinue his trek, but not necessarily where he left off
fully
forward toward a miniature whirling discotheque
the year before. Akira's journey was noncontiguous.
light
flashing red and blue. He watched and looked,
Stretches were completed here and there. We believe
cocking his head to catch the glint off
that he was nearly finished with this
the
tiny spherical dancers reflecting
walk of a lifetime, left only with the
This tribute
off
those
red and blue lasers. "Here
long stretch across the central plains describes only a fragment
they
are,
Akira! Those pesky zooa journey he m a y undertake now.
o
f
a
life
w
e
l
l
l
i
v
e
d
.
plankton,
swarming
as you modeled
What did he do on the last leg of his
in
1984
and
as
y
o
u
imagined
20 years
adventure, this one of so many? After
ago
w
h
e
n
you
watched
those
midges
in
the
cornfield."
completing work with colleagues in Cornell, Akira
The blue light caused a dramatic display: wild little
walked from the Finger Lakes of upstate N e w York to
cladocerans ran out of their corner of the vessel to line
the shores of Lake Michigan. Rudi Strickler and I then
up on the laser beam. Akira was captivated, just like
met him at the train station. We sped him through the
those little plankters.
streets of Milwaukee, d r o p p e d off his gear and - with
"Can you set u p another laser?" Akira asked. "I
our o w n multifaceted gring - led him into the darkened
w o u l d like to be able to change the distance between the
laboratory of the Center for Great Lakes Studies at the
swarm markers to see if these animals can demonstrate
University of Wisconsin.
"
30
\.
e/ 7
Oceanography
•
Vol. 12
" No. 1/1999
the effects of another strange attractor." Rudi and I
looked at each other, knowing what a great time we
would have working with Akira on this joint project: to
study swarming by zooplankton supported by the
National Science Foundation (Figures 15 and 16).
( [?,?..,ia,
,,
M a t h e m a t i c a l m o d e l s for s w a r m m a i n t e n a n c e
(NSF p r o p o s a l c o n t r i b u t i o n written by Akira
O k u b o , f u n d e d f r o m 1993 - 1997)
)
/
,/
S_~'t"- - ~
JI
~"J
INSECT S W A R M I N G
?~'J";%Ft.St4
seos,t.~•
. . . . .
?
]..
e,IG
x
x
8,11~. e ~ I~'~'''~,'.
v';,~
,."
",,...d
+
A iac~Pe,- l L
-4;
/0
/
~,d'*
J
td"
/
StzE
Io a
/~*
( L ~,,,)
OK~l~o (iqS't~
O ~ , 6 o . H;f¢k~dl
Figure 16. Reproduction of an overhead transparency drawn by Akira
showing the relationship between body size and Reynolds ;mmber appropriate for swimming motions across taxa and kingdoms, from the small
(bacteria) to large (whales)scale.
Excerpts from reviews of our grant proposal show
how strong the appreciation was for this great man:
"Okubo is internationally one of the top level scientists
working in the area of dynamics of biological organisms
and biological oceanography...Okubo is a well-established leader in the field of analysis of animal
motion...Dr. Okubo is a long-standing expert in the
topic of swarming behavior. Indeed, one could say
without any hesitation that he is one of the founders of
this whole fascinating line of research."
Following is the fragment of our NSF proposal that
Akira Okubo had been working on when he passed
Oceanography o VoL I 2 • No. 1/1999
away. Not only is it clear to the reader the love he had for
teaching, but he obviously liked to work in a team, collaborating with experts in other topics. As disciples of
this great mentor, we continue to strive to work together in the style of a man whose own style was inimitable.
A central question of aggregative behavior is how a
swarm is maintained despite apparent randomness in
the motion of swarming animals so as to balance the
tendency of spreading by random motion. Analyses of
insect swarming behavior, recorded on movie fihn by
Chiang (1968) addressed the objective of determining
how long an insect (midge, Anarete prichardi Kim)
stays in a swarm.
By tracking individual insects and analyzing their
movements in several segments of the fihn, a number of
statistical characteristics of swarming individuals were
calculated, including the mean, variance, standard
deviation, skewness, and kurtosis of the insect coordinates; insect number-density distribution in space,
frequency distribution of insect velocities, velocity
autocorrelations, and diffusivity associated with insect
random motion (Okubo and Chiang, 1974).
The results were enlightening ;tot only in the kinematical aspect of swarming but also in the biological
interpretations of swarming behavior. The same basic
data also were used to analyze the acceleration and force
fields of midges in swarming. The existence of the deterministic attractive force was proven (Okubo et al.,
1976). Further detailed behavior of mid,~e swarmin~
permitted the construction of a simple mathematical
model for the dynamics of swarming (Chiang et aI.,
1978; Chiang et aI., 1980; Goldsmith et al., 1980;
Okubo et al., 1980; Okubo, 1986).
Hence, a swarm is maintained by the balance
between deterministic and stochastic forces experienced
by swarming individuals. Once a swarm forms, it
remains intact for some time with little change in its
spatial dimensions, constituting a "quasi-stationary
aggregation." The stochastic force alone makes a swarm
spread out to occupy a larger area or volume as time
goes on where this tendency of spreading is simply a
consequence of diffusion.
To adequately model swarming, therefi~re, it will be
necessary to inhvduce certain regularit&s in motion
superimposed upon the random motion. From the analogy to insect swarming, the nature of the regular, deterministic force is expected to be attractive toward the
swarm center. Detailed tracking data of zooplankton
swarming behavior will enable us to evaluate the fi?rce
field as a.hmction of distance from the swarm center.
3I
Also the random component of the movement of the
swarming animal may be estimated from the trajectory
of the animal.
As a statistical measure of the swarm dimension, the
variance of the animal displacement will be calculated by
averaging the squares of the distance from the center over
a sufficiently large number of individuals in a swarm.
Kinenzatically the variance is related to the velocity autocorrelation of the animal motion. We predict that the
velocity autocorrelation should oscillate about zero in
such a manner that a long term average approaches zero.
This behavior of the velocity autocorrelation characterizes swarming that is distinct from diffi~sion. The variance, i.e. the spatial size of swarm, can be estimated from
the velocity autocorrelation. This variance model will be
tested with experiments through the determination of the
velocity autocorrelation of zooplankters in a swarm.
We also zuill consider a dynamical model for swarm
maintenance. Newton's equation of motion will be
applied to swarming animal motion. Four main forces
are assumed.
The net effect will produce a more or less uniform
density in the central region of swarms. Thus, the zooplankton spatial distribution is predicted to be
platykurtic. These theoretical predictions will be tested
against data. If the comparison is favorable, we then
will proceed solving the nonlinear model equation
numerically to obtain more exact solutions.
This excerpt from the NSF proposal illustrates his
thinking about the way mathematics can be used to
describe the dynamic interactions between physics and
biology in zooplankton behavior. Unfortunately, Akira
did not live to complete this research project.
Akira Okubo was a man whose innate curiosity made
him so great. Can you just see him sitting in the middle
of a cornfield patiently waiting and watching insects as
they swarm? One time, I asked Akira to help me: "Do
you want to go and collect copepods with me?" Akira
said, "Ooo, nov, no. I get seasick!" I replied, "Well, don't
worry - we won't be going out on a boat!"
So I took Akira and our plankton net over to Stony
Brook Harbor, near to both his home and the historic
1. The frictional force or drag of zooplankton as they
Three Village Inn where many Marine Sciences
swim in water.
Research Center functions were held. It was a warm
spring evening. He lowered the net into the water and
2. The deterministic force of an attractive nature which
pulled
it along the sea wall. "Ehhh," Akira said, "this is
is zoorking toward the center of swarm and depends
like
walking
the dog." We walked and talked and
not only upon the distance from the center but also
watched
the
sun
set with red clouds over Long Island
upon the density of conspeci)qc members.
Sound. It was so much fun working with Akira. ! tried
3. The stochastic force due to randomness of zooplankto show him everything we saw copepods do. I would
ton motion.
show him the behavior, and he would respond with
mathematical equations.
4. The gravitational force arising from a higher densiTeaching in the classroom with Akira was always
ty of zooplankton body than the density of the surinspiring. His logic was impressive. Even my old letters
rounding water.
from him were often written in a different language,
An approximate analytical solution of the dynamical
that of mathematics, depicted in symbolic logic.
model equation may be obtained by the method of equivStarting in one dimension, he made a point, then
alent linearization (Bulsara et al., 1982). From the anaextended it to the line of thought, adding depth with a
lytical solution we can calculate the velocity atttocorreknowledge that was fathomless.
lation, the variance of animal displacements, speed freHe studied the two-dimensionality of swarming of
quency distribution of swarming anithe water skater, Halobates [T. Ikawa,
mals, among other kinematical characTeaching in the classroom
this volume]. He solved the three
teristics. These theoretical results will with Akira zuas always inspiring. dimensional spatial distribution of aerbe compared with observed data.
ial insects by tracking two-dimensionHis logic was impressive.
Assuming the stochastic force is qfi
al shadows [Alan Elliott, this volume].
white noise type, we can formulate a
When Akira first went into the
Fokker-Planck equation for the probability density funchospital in 1995, I brought a videotape to entertain him
tion of velocity and displacement of the swarming aniand keep his mind on different thoughts: the tape was
mal. The solution of the Fokker-Planck equation enables
on mate-tracking by swarming copepods, where the
us to obtain the theoretical distribution qf velocity (and
male chases the female! That got him thinking... I met his
speed) and the spatial distribution of swarming animals.
colleague, Dr. Chiang, who was visiting Akira that day
It is conceivable that the deterministic attractive force is
too. This new movie was so exciting to him. Akira could
a manifestation of density dependent advective velocity
now fuse his keen interests in diffusion and swarming,
toward the swarm center. It is a built-in mechanism to
and mathematically model the use of pheromonal trails
maintain a sharp boundary of concentration despite a
for mate-seeking in pairs of copepods.
y,eneral tendency to spread by the random component of
motion.
32
Oceanography • VoL 12 • No. 1/1999
N e w s Item: N e w E d i t i o n of O k u b o ' s
Classic Text to A p p e a r in 2000
Simon A. Levin
Department of Ecology and Evolutionary Biology
Princeton University
Princeton, New Jersey, USA
Figure 17. Jeamu'tte Yell and Akira at Hana's Restaurant, Port
Jefferson, Long Island, New York, celebratingAkira's 70th birthday, 5 February 1995. The fresh flower leis were brought from
Hawaii by Jeannette to celebratethe occasion (photoby Jerry,Liu).
A n d n o w - while he is w a n d e r i n g the Great Plains,
his smile for us only a m e m o r y b u t his thoughts still so
alive, we mortals struggle to fit nature to his equations
because gone is the m a n w h o fit equations to nature.
REFERENCES
Bulsara, A., K. Lindenberg, and K.E. Schuler, 1982:
Application of linearization methods to driven nonlinear
systems. In: Instabilities, Bifurcations, and Fluctuations in
Chemical Systems. L.E. Reichl and W.C. Schieve, eds.,
University of Texas Press, Austin, Texas, 400-410.
Chiang, H.C., 1968: Ecology of insect swarms. V. Movement of
individual midges, Anarete pritchardi, within a swarm.
Annals of Entomol. Soc. Amer., 61, 584-587.
Chiang, H.C., B.J. Metfler, A. Okubo,and A.S. Robbins, 1978:
Coupling of midge individuals in a swarm, Anarete pritchardi (Diptera: Cecidomyiidae). Annals of Entomol. Soc. Amer.,
71,859-861.
Chiang, H.C., A. Goldsmith, and A. Okubo, 1980: Interaction
of male and female midges, Anarete pritchardi Kim, leading
to coupling. Annals of EntomoL Soc. Amer., 73, 504-513.
Goldsmith, A., H.C. Chiang, and A. Okubo, 1980: Turning
motion of individual midges, Anarete pritchardi, in swarms.
Annals of Entomol. Soc. Amer., 73, 526-528.
Okubo, A., 1980: Diffusion and ecological problems: Mathematical
models. Springer-Verlag, 254 pp.
Okubo, A., 1984: Critical patch size for plankton and patchiness. In: Mathematical Ecology Lecture Notes in
Biomathematics, Vol. 54, Springer-Verlag, 456-47.
Okubo, A., 1986: Dynamical aspects of animal grouping:
swarms, schools, flocks, and herds. Advances in Biophysics,
22, 1-94.
Okubo, A. and H.C. Chiang, 1974: An analysis of the kinematics of swarming of Anarete prichardi Kim. Res. PopuI. Ecol.,
16, 1-42.
Okubo, A., H. C. Chiang, and C. C. Ebbesmeyer, 1976:
Acceleration field of individual midges, Anarete pritchardi
(Diptera: Cecidomyiidae), within a swarm. Canadian
Entomol., 109, 149-156.
Okubo, A., D.J. Bray, and H.C. Chiang, 1980: Use of shadows
for studying the three dimensional structure of insect
swarms. Annals of Entomol. Soc. Amer., 74, 48-50.
Okubo, A. and J.J. Anderson, 1984: Mathematical models for
zooplankton swarms: their formation and maintenance. In:
The Oceanography Report. EOS, American Geophysical
Union, 731-733.
Oceonogrophy • Vol. 12 • No. 1/1999
Akira Okubo's elegant m o n o g r a p h , Diff~lsion and
Ecological Problems: Mathematical Models (SpringerVerlag, 1980), is a classic of the literature. I take some
pride in it, because it came about in part because of m y
inability to read Japanese. Akira had presented me with
an earlier version of the work, but I could u n d e r s t a n d
no more than the mathematical equations that were surr o u n d e d by Japanese prose. It was clear that I n e e d e d to
k n o w the contents of the book; hence, I took advantage
of m y role as Editor of the Springer Biomathematics
Series to invite Akira to contribute a new edition, this
time in English. As each chapter arrived on m y desk for
comments, it was clear that I had hit a gold mine. The
rest is history.
Akira's book stimulated a t r e m e n d o u s a m o u n t of
work in applying diffusion models to ecological problems. The subject has advanced a great deal in the nearly two decades since its publication. Hence, a few years
ago, I went back to Akira and encouraged him to consider a new and u p d a t e d version. Unfortunately, he did
not live long e n o u g h to complete it, but he left copious
notes about what he intended to do. His friend, Keiko
Parker, sent me those notes, with Akira's wish that I see
them through to publication.
Regrettably, Akira's s y m p h o n y was unfinished, so it
is left to his students to fill in the details as best they can.
Springer-Verlag was eager to publish an update, and
Akira's admirers and disciples have stepped forward.
Each chapter will be revised b y one or more of Akira's
colleagues, u p d a t i n g the old material in the light of
Akira's written blueprint and e x p a n d e d as appropriate.
The v o l u m e promises to continue Akira's legacy, and
will appear in early 2000. Editing the collection will be
a labor of love for me, a last o p p o r t u n i t y to honor Akira,
w h o cannot be replaced.
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